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Photon asymmetry measurement in radiative muon capture on calcium-40 Pouladdej, Ali 1987

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PHOTON ASYMMETRY MEASUREMENT IN RADIATIVE M U O N C A P T U R E ON CALCIUM-40 By ALI  POULADDEJ  B.Sc, Manchester University, 1980 M.Sc, London University, 1982 B.Sc, Nottingham University, 1984  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE  in T H E FACULTY OF G R A D U A T E STUDIES Department of Physics  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A August 1987 © Ali Pouladdej, 1987  In  presenting  degree  at  this  the  thesis  in  partial fulfilment  of  University of  British Columbia,  I agree  freely available for reference copying  of  department publication  this or of  and study.  this  his  or  her  that the  representatives.  may be It  is  thesis for financial gain shall not be  permission.  Department The University of British Columbia 1956 Main Mall Vancouver, Canada  DE-6(3/81)  requirements  I further agree  thesis for scholarly purposes by  the  for  an  advanced  Library shall make it  that permission  for extensive  granted  head  by the  understood  that  allowed without  of  my  copying  or  my written  Abstract  A p h o t o n a s y m m e t r y measurement i n radiative m u o n capture ( R M C ) o n  4 0  C a has  been carried out i n order to determine a value for the induced-pseudoscalar coupling constant (gp)', thereby allowing us to investigate the possible renormalisation effect for this c o u p l i n g constant inside nuclear matter. Negative muons from the M 2 0 A channel at T R I U M F were stopped i n a target.  4 0  Ca  T h e resulting R M C photons were then converted by a 5 c m thick N a l  converter placed i m m e d i a t e l y i n front of our m a i n detector, a N a l crystal of size 46 c m 0 x 5 1 c m . T h e purpose of the converter was to provide d i s c r i m i n a t i o n against the prolific h i g h energy neutron background. F r o m a fit to our t i m e spectrum for the high-energy (57 M e V < E  1  <95 M e V )  photons, we were able to extract a value for the R M C photon a s y m m e t r y ( » ) . T h i s 7  t i m e spectrum, w i t h 5200 "clean" photons a n d a signal to noise ratio of ~ 7 , showed the p h o t o n asymmetry quite clearly. W i t h our measured value for the p h o t o n asymmetry: a predicted  value  of  the  induced-pseudoscalar  coupling  7  = 1.0010.23, the  constant  becomes:  gp = (0.0te'°) gA', this range contains the (theoretical) unrenormalised value of Goldberger-Treiman: gp « 7gA-  n  Table of Contents Abstract  ii  List of Tables  v  List of Figures  vi  Acknowledgements  viii  Dedications I  II  ix  Introduction  1  1.1  Weak Interactions  1  1.2  Motivation for Studying RMC on C a  3  4 0  Theory  7  II. 1 Depolarisation Effects Associated with Muon Beams  8  11.2 Free Muon Decay  9  11.3 The Formation and Evolution of Muonic Atoms  12  11.4 Bound Muon Decay  13  11.5 Nuclear Muon Capture and the RMC  17  11.5.1 Nuclear Muon Capture and its Theoretical Treatment  . . .  17  11.5.2 RMC Within the Standard IA  20  11.5.3 Beyond the IA  23  11.5.4 The Physically Measurable Observable, cv  24  11.5.5 Possible Renormalisation of gp in Nuclear Medium  27  7  III Description of the Experiment  31  III.l Beam production and the Muon Channel iii  31  111.2 Experimental Geometry, Technique and Philosophy  35  111.3 The Detectors  39  111.4 Design of the Experiment  43  111.4.1 Target Design  43  111.4.2 Shielding and Background Considerations  46  111.5 Electronics and Data Acquisition  IV Data Analysis  51  61  IV. 1 Introduction  61  IV.2 Choice of the Fitting Function .  63  IV.3 Calibrations  64  IV.3.1 Time Calibrations  65  IV.3.2 Energy Calibrations  66  IV.4 Discussion of the (NEU-related) Cuts and the Energy Reconstruction 71 IV. 5 Neutral Data Analysis  79  IV.5.1 Background Term  79  IV.5.2 Estimation of the Precession Frequency and the Initial Phase 80 IV.5.3 Extraction of the Observed Photon Asymmetry  V  Discussion of Results, and Conclusions  84  95  V. l  Brief Review of Some Aspects of Data Analysis  95  V.2  Determination of a  96  V.3  Extraction of gp/gA  98  V.4  Conclusions  y  102  Bibliography  104  iv  List of Tables I  Normalised decay rates /N -(E)dE, integrated asymmetry coefficient / A -(E)dE and their ratios  17  The energy-averaged photon asymmetry as calculated from several different models  25  III  Plastic scintillation counters dimensions  40  IV  Target inclination optimisation results  46  V  Optimisation results for the "All Three" (AT) option of LOw (LO) cut  77  e  e  II  VI  Optimisation results for the "All Three" (AT) option of LOw (LO) cut  77  VII  Results of the time scan for the e  VIII  Results of the energy scan for the e time spectrum  IX  ^u~stop statistics for the RMC runs  84  X  Optimisation results for  85  XI  Table of the time scans for the observed photon asymmetry  XII  Final PR and SF  XIII  Summary of photon asymmetry measurements on C a  XIV  Values of gp/gA as determined from OMC experiments on hydrogen. 101  XV  Values of gp/gA nuclei  +  time spectrum  82 84  +  Ei  max  . . .  fits  94 4 0  a  s  93  97  extracted from OMC experiments on complex 102  v  List of Figures 1  Point interaction of four fermions  5  2  Pion propagator enhancement diagram  5  3  Energy spectrum for electrons resulting from the free decay of muon.  10  4  Theoretical energy-dependent asymmetry for free decay of fi .  11  5  Theoretical decay-electron spectra for \i~ in lead and iron  15  6  Theoretical A - for bound decay of \T  16  7  The Feynman diagrams employed by Rood and Tolhoek for RMC amplitude evaluation. .  21  8  The energy-averaged photon asymmetry as a function of gp/gA-  26  9  The layout of the M20 channel  33  10  Simplified M20 channel  34  11  The experimental geometry.  36  12  The gas target  39  13  Target geometry.  45  14  Detailed shielding plan in the vicinity of M20A beamline  49  15  Electronics for processing the TINA and the BARS signals  55  16  The trigger logic  57  17  Schematics of the pile-up detection electronics  60  18  The "ideal" energy spectrum for 7r~p reaction at rest  68  19  Typical TINA observed energy spectrum  68  20  The position dependence of the energy function G;  70  21  The corrected energy function GC;  71  22  Raw energy spectrum for one member of LST  72  23  Minimum energy deposition spectrum in LST for a given event. . . .  76  24  LO cut optimisation for LST  78  25  e  +  . . .  e  +  • • •  time spectrum for the energy slice 15.0 MeV < E + < 55.0 MeV. e  vi  81  26  The experimental energy spectrum for the decay e data  81  27  Stability of the (integral) observed e  83  28  Final neutral time spectrum shown at the collected 1.011 nsec time resolution  86  29  Neutral time spectrum with the time resolution of 43.47 nsec  89  30  Attenuated observed photon asymmetry Aj* )  89  31  Neutral time spectrum with the time resolution of 10.11 nsec  90  32  Attenuated observed photon asymmetry A^ ^  90  33  Time scan of the attenuated observed photon asymmetry  91  34  Comparison of the time scans for the observed photon asymmetry  35  Physical photon asymmetry as a function of the induced pseudoscalar coupling constant  +  +  asymmetry  43  10  vn  .  92 99  Acknowledgements First of all I would like to acknowledge the cumulative efforts invested by several people over the years in the original RMC experiment at TRIUMF (E47) from which this present work (E364) has benefitted greatly. I am grateful to all of the following people who participated in this RMC experiment: Drs. G. Azuelos, A. Burnham, T. Gorringe, M.D. HasinofF, D. Horvath, A.J. Larabee, C.J. Virtue and D. Wright; and T. Noble. I would like to extend my gratitude to my supervisor professor M.D. HasinofF for having made this project a true learning experience in so many ways. I am grateful to him for his constant guidance throughout the entire course of this work. Dr. C.J. Virtue is thanked for many useful and fruitful discussions. Finally, I would like to thank professor D.F. Measday for his useful comments and critical review of the final draft of this thesis. The research reported here was supported in part by the NSERC.  Vlll  Dedications  With great love to my parents.  ix  Chapter I Introduction 1.1 W e a k I n t e r a c t i o n s The interest in the weak interactions goes back to the early days of radioactivity where the weak force manifested itself through a common form of radioactivity, the /?-decay of nuclei: n -> p + e~ + v  e  (1)  The establishment of the weak force as a separate and independent interaction in the nature was rather gradual. This identification came only after observing other weak reactions such as the //-decay and //-capture etc., which possessed approximately the same "coupling constants" as that for a /?-decay; a phenomenological observation which forms the basis for the "//-e universality." Weak interactions can be categorised into three classes: Purely leptonic, semileptonic (with subclasses: Strangeness changing and strangeness conserving), and non-leptonic, based on the proportion of the leptons present in the reacion. For example, the //"-capture process \J>~ + p  n+  (2)  is an instance of a strangeness conserving semi-leptonic reaction; in the context of nuclear capture (say in Ca), this is referred to as Ordinary Muon Capture (OMC): 40  i-+ Ca^v + K* 40  f  40  fi  (3)  This work is concerned with a much less probable version of the above nuclear 1  process known as Radiative Muon Capture (RMC): /  x  -  +  4 0  C  a  ^ ^  +  T  +  4 0  The Branching Ratio (BR) ARMC/^OMC ~ 2 x l 0  - 5  K  *  (  4  )  makes RMC much less favourable  and its observation more difficult. Fermi was the first person to propose quantitatively (based on a point interaction of the four fermions, in relation to nuclear /?-decay), a meaningful theory of the weak interactions [1] in terms of vector (weak) currents; this is now commonly known as the Universal Fermi Interaction (UFI). Guided by the form of the electromagnetic (em) interaction, he proposed a current-current type of coupling for the weak interaction. Fermi's inspired guess of a Vector-Vector form of the weak amplitude was truly remarkable as he had made a very specific choice from amongst many Lorentz-invariant amplitudes available to him. Fermi's original V form preserved parity; but with the discovery of the parity violation [2] this had to be changed. The implementation of parity non-conservation was quite straightforward and amounted to replacing 7 by 7 ( 1 + 7 5 ) in the formam  M  tion of the weak currents (the projection operator (1 + 7 5 ) allows only left-handed leptons and provides the required parity violation.) This meant that weak currents were now V—A (Vector—Axial vector). Furthermore, the V—A choice now accounted for both the Fermi (F) (AI= 0, where I=nuclear spin) and the GamowTeller (GT) (AI= 0, t l ) type of nuclear transitions. This modified form of V—A theory in which the odd and even parity amplitudes have roughly the same magnitude constitutes what is known as the "principle of maximal parity violation." In the case of RMC for a polarised yT, the parity violation is manifested as an angular correlation between the direction of the emission of the RMC photon and direction of the polarisation (spin-quantisation axis) of the fi~ (let this angle 2  be 9). T h e n , the observed angular d i s t r i b u t i o n (the " Y i e l d " ) for R M C photons is Y (9) y  = l + ({P -)a )cos9 tl  (5)  1  where cv is the "energy-averaged" or the "physical" p h o t o n a s y m m e t r y a n d ( P - ) 7  M  is the average p~ (spin) polarisation before capture.  T h e V — A theory predicts  100% right circularly polarised photons, a n d so for a pure V — A theory cv = 7  +1  exactly.  1.2  Motivation for Studying R M C on  4 0  Ca  T h e weak interaction H a m i l t o n i a n for m u o n capture i n terms of the four fermion c o u p l i n g a n d the G e l l - M a n n - C a b i b b o universality [3,4] is given by Hw =  MVx  + A )rP x  P  ^„  7 A  (1 +  7  ) ^ + h.c.  5  (6)  where G is the F e r m i coupling constant a n d 9 is the C a b i b b o angle; n u m e r i c a l l y Gcos9 = (1.4122010.00043) x 1 0  - 4 9  erg c m . T h e phenomenological form of the 3  hadronic vector a n d a x i a l vector are  V\ = gvl\  A\  - ^(9M/m)a q Xp  p  = SUTATS - (igp/m^qwa  - (igs/mjqx  (7)  - ^(gT/m)<T q y Xp  p  (8)  5  where 7>'s are the D i r a c matrices w i t h 75 = 71727374 a n d | = I, the cr-matrices are 7  defined as a\  p  — |(7A7/> Iplx)—  m  a n d m ^ refer to the nucleon a n d the m u o n masses  respectively. T h e 4-momentum transfer is given by q\ — n\ — p\, where n\ a n d p\ are the n e u t r o n a n d the p r o t o n 4-momenta respectively. T h e parameters (gv, gM, and g ) s  a n d (gA, gp, a n d gx) are the basic coupling constants (or less commonly,  "form factors") for the semi-leptonic interaction; they are functions of q . 2  The  subscripts: ( V , M , a n d S) a n d ( A , P , a n d T ) refer to (Vector, weak M a g n e t i s m , a n d Scalar) a n d ( A x i a l vector, Pseudoscalar, a n d Tensor) interactions respectively. 3  The V>'s are the spinor functions for the fermions a. The coupling constants g\{, a  gs, gp, and gr are "induced" by the strong interaction of the hadrons. The induced-pseudoscalar coupling constant (gp) forms the subject of the investigation in this thesis. The main component for this coupling is the one pion exchange diagram with a propagator  m 2  \ 2 q  •  In O M C , the 4-momentum transfer is  fixed at q w 0.9m , while in the case of R M C a value of q « — m is achievable 2  2  2  2  (near the maximum photon energy); thereby enhancing the amplitude of this diagram by a factor of ~ 3.5. It is interesting to point out that because of the factor q which multiplies the pseudoscalar coupling in the Hamiltonian, this coupling is absent in the /?-decay. In summary we note that the pseudoscalar coupling which is absent in /?-decay and affects the O M C very little, strongly influences the high energy part of the photon energy spectrum and the absolute rate for the R M C . It was the identification of the strong dependence on gp of the total rate for the photon energy spectrum which initially motivated R M C studies. The very low rate for R M C  (RA:>57  ~ 2x 10 ) favoured rate measurements on heavy nuclei because for -5  these nuclei, the coulomb attraction concentrates the muon's wavefunction over the nuclear volume, thereby enhancing the capture rate with respect to the muon decay. However, with heavier nuclei many theoretical difficulties follow as a matter of course because of the uncertainties of the nuclear states involved in the interaction. To reduce such effects, doubly magic nuclei such as O and Ca ie  40  with relatively  simple nuclear structures became favoured candidates for theoretical investigations, and these nuclei were consequently chosen as experimental targets. The importance of the muon-spin photon angular correlation has been recognised already, and there is a gradual shift in favour of the photon asymmetry measurement; the reasons for this being two fold. Firstly, the energy-dependent photon asymmetry coefficient T (fc) which describes the angular distribution of the pho7  4  Figure 1: Point interaction of four fermions.  Figure 2: Pion propagator enhancement diagram. 5  tons is strongly dependent on gp [5]. Secondly, T (k) has been shown [6,5] to be 1  almost insensitive to the details of the nuclear structure model employed in the calculations; this makes a an almost model-independent observable for RMC. A 7  measurement of o: therefore provides a direct method for determining gp for (i~ 7  inside the nucleus. The Partially Conserved Axial Current (PCAC) formulation allows an estimation of gp for a "free" nucleon. This value:  gp  « 7<JU, is known as the Goldberger-  Treiman value (GTV henceforth) [7]. By comparing the experimentally determined value of gp with the GTV it is then, at least in principle, possible to draw conclusions about the renormalisation of gp within nuclear matter [8]. Finally, it has also been pointed out [9] that the value of gp inside the nucleus might provide some information about the nuclear polarisability (a).  6  C h a p t e r II Theory  In this chapter the manner in which the muon interacts with matter will be examined. Within this framework, the characteristics associated with both the free and bound decays of the muon will be considered; special emphasis is placed on the Radiative Muon Capture branch of the bound decay. The asymmetry observed in the emission of RMC photons is related to the parity non-conservation in weak interactions. In RMC asymmetry measurements (employing polarised muon beams), a quantity of great importance is the \x~ polarisation before capture, i.e.  the \x~ residual polarisation ((P^-)); it serves  as the normalisation factor for the observed photon asymmetry. Consequently, this prompts an interest in the study of depolaristaion effects associated with muon beams. In general, the two distinct sources of depolarisation are kinematic depolarisation and depolarisation in atomic capture and cascade. The former affects pT and /x beams equally, while the latter is of relevance to \i~ beams only; this is due +  to the fact that yT and fi  +  behave quite differently in matter.  Upon entering the target, a fj, thermalises within ~ 10 — I O +  -9  - 1 0  sec, primarily  by causing ionisation in the medium; it eventually (in the case of a metallic target such as Ca) occupies an interstitial or defect site in the crystal lattice. In the 40  slowing down process, a p loses hardly any of its polarisation. Finally, it decays as +  an essentially free muon with the characteristic life time of ~ 2.2 yusec. In contrast to this, a \x~ suffers considerable depolarisation during its cascading down to the K-shell of the muonic atom (cascade time to the ls orbital for p~ after its atomic  7  capture is ~ I O  - 1 3  sec); retaining only ~ | of its initial (i.e. before entering the  target) polarisation. It then either gets captured by the nucleus or decays as a bound  muon; the rates for these have to be added to obtain the total disappearance  rate; i.e. the apparent life time is reduced. (r - in C a is ~333.0 nsec.) 4 0  M  I I .1 D e p o l a r i s a t i o n E f f e c t s A s s o c i a t e d w i t h M u o n  Beams  In this section, the two sources of depolarisation affecting muon beams are briefly discussed. Kinematic depolarisation arises essentially due to the method of production and collection of muons. Collimated muons are derived from the decay of slow moving pions. Even though the muons are completely longitudinally polarised in the rest frame of the pion, there will be kinematical depolarisation in the lab frame due to the Lorentz transformation and because backward and forward muons have opposite polarisations.  Additionally, the magnets used in collecting the muons  further affect the polarisation. This source of depolarisation affects uT and u.  +  beams in the same manner. The depolarisation in atomic capture and cascade, however, concerns beams only. During the brief time taken for the /i~  fi~  to slow down depolarisation is  negligible. In the process of atomic capture into high angular momentum states, most of the fi~  polarisation is lost; it retains only ~ | of its initial value. Then  follows the atomic capture down to ls orbital which results in a further 50% loss in polarisation. For nuclei with spin, there is still an additional source of depolarisation due to spin-spin interaction; for C a with zero nuclear spin, this latter source 4 0  of depolarisation is absent. 8  II.2  Free M u o n Decay  The free muon decay scheme with 98.6% branching ratio is (9)  p, -» e + u + +  +  e  p~ —• e~ + u + Up e  (10)  where the law of conservation of leptons requires the emission of a neutrino and an antineutrino. The electrons (positrons) so emitted have a maximum energy of E  e  52.8 MeV.  =  For leptonic decays, the normalised electron distribution given by the two component neutrino theory is N(x)  where  x = E /E e  e  =  4z  2  3(l-x)  + ^p(4x-3)  (11)  and the electron rest mass has been neglected. The shape of  the spectrum depends only on the Michel parameter (pi) [10] which is expressible in terms of all the coupling constants involved in the interaction. If the neutrinos satisfy a two-component theory then p is calculated to be | ; independent of whether the /i-decay interaction consists of (V,A) or (S,P) coupling. The energy distribution in this case becomes N(x)  = ( 6x  2  - Ax ) 3  (12)  where dE  e  ^  ^ ^ 3  This is shown in figure 3. Radiative corrections [11,12] will slightly modify this electron spectrum and result in a value of p somewhat different from | . Experimentally measured values of p [13] are found to cluster about p = 0.75. 9  0.0  0.2  0.4 x=E  0.6 J52.8  0.8  1.0  Figure 3: Energy spectrum for electrons resulting from the free decay of muon; this is a Michel spectrum with p = |.  Since parity is not conserved in p- decay, there will be an asymmetry in the angular distribution of the decay electrons (positrons). For polarised muons at rest, the theoretical angular distribution is given by [14] dN(x, 9) = ^TTX [ (3 - 2x) t C (1 - 2x)cos0 ] dx dn  (14)  2  for p,-, where p =the electron momentum, x= p /p e  e  e  ie. the momentum normalised  to its maximum value, and 9 is the angle between the direction of the electron's momentum and the spin of the decaying muon. For the two component neutrino theory > C  + 9A  9v 9A =  A  + A  9y (  1  5  )  for which the V—A interaction yields a value of ( = —1. (Applying radiative corrections gives £ = —1.003 [12].) 10  0.4 0.6 x=p /52.8 e+  Figure 4: Theoretical energy-dependent asymmetry for free decay of .  The decay electron angular distribution is W (0) = (1 + A cos6) e  (16)  e  and the observed angular asymmetry (taking the solid angle for observation into account) for / i - decay becomes (17)  1(d) d6 = (l + A cos6 ) sinO « ( 1 + ( ^ — ^ - c o s 6 ) sin8 dd e  V  (  3  ~  2 x  )  J  The energy dependent asymmetry parameter is then A (x)= +  tC  (18)  (3 - 2x)  Figure 4 shows A +(a;) for the free decay of fj, . From the above equation it is seen +  e  that at the maximum energy A + (1) = +£ and at zero energy A + (0) = t\(e—  For the  e—  integrated spectrum (resulting in an energy-averaged asymmetry) ^A + ^ = which for £ = — 1 gives (A + ) = + | and (A - ) = —|. 11 e  e  II.3  T h e Formation and Evolution of Muonic Atoms  The slowing down process of /x in matter and its atomic capture was first investi-  gated by Fermi and Teller [15]. Through simple arguments the authors show that that the atomic capture rate (A ) is proportional to the nuclear charge (Z): a  A  oc  0  (19)  Z  with the atomic capture occuring after the muon had lost almost all of its kinetic energy. This is now known as the Fermi-Teller Z-law for the atomic capture of muons. Deviations from this law are generally found to be due to the atomic shell structure and chemical environment associated with the stopping medium [16]. The slowing down and entrapment of the aT proceeds through several stages [17]. • High to low (few keV) energy: Muons enter the target with energies of the order of tens of MeV, and possess velocities larger than those of the valence electrons. They lose most of their energy via collisions with atomic electrons and eventually achieve velocities comparable to those of the valence electrons. The time scale involved for this step in condensed matter is ~ 10 — I O -9  - 1 0  sec.  • Atomic capture and electromagnetic cascade: Once a muon has lost most of its kinetic energy, it gets captured by a host atom into a high orbital angular momentum state, thereby forming a muonic atom. Following its capture, the yT will enter (within ~ 1 0 principal quantum number  w  -14  sec) a muon atomic orbit which has a  (m^/m )2 e  R2 14. Now, since all of the low-  lying muonic states are empty, the muon will cascade down to the lowest available quantum state, ls. The energy loss in this step results from Auger and radiative transitions. The former is important for large n^, while the radiative transition (overwhelmingly of the electric dipole type) dominates for lower values of n^. The cascade time is ~ 10~ 12  13  sec.  • Disappearance of the muon: Once in the ls orbital, the muon either decays or is captured by the nucleus.  B o u n d M u o n Decay  II.4  The Z value of the nucleus affects the disappearance rate and hence the life time of the bound muon; Z also dictates which of the processes, bound decay or nuclear capture, dominates. In hydrogen, the capture to decay probability ratio is ~ 10~ . This ratio increases for larger values of Z such that for Z ~ l l the capture 3  rate becomes comparable to the decay rate. Eventually at even higher Z values, the capture process begins to dominate. For Z~50 the ratio of capture to decay probabilities is ~25. The removal or disappearance rate (A ) for a bound muon in the ls orbital is r  the sum of the bound decay rate ( A ^ ) and the nuclear capture rate (A ) oW  c  A  = A  r  b ound d  + A  (20)  c  The life time of the bound y~ is then given by the inverse of A : r  For the case of yC in C a , r - ~ 333.0 nsec [18]. 40  M  Porter and Primakoff [19] were the first to point out that the decay probability for free  (h.{ ) ree  and bound decay (A J ) b  mnd  differed. Later investigations of this effect  involved various approximations for muon and electron wave functions; the earliest notable effort was undertaken by Gilinsky and Mathews [20]. In their calculations they represented the muon by a relativistic wavefunction for a point nucleus, and the outcoming electron by a Sommerfeld-Maue [21] wavefunction. For this work, electron mass and nuclear recoil effects were ignored. The calculations by Hanggi et al. [22] aimed at removing the aforementioned approximations. These authors calculated the energy spectrum for the emitted 13  electrons by employing accurate electron and muon wavefunctions generated by the numerical integration of the Dirac equation involving a potential which included vacuum polarisation. Furthermore, they accounted for the non-zero mass of electron and the nuclear recoil effects. In this work, the bound decay effects are accounted for in terms of the Gilinsky and Mathews model. Qualitatively, the effective mass of a muon in the ls orbital is reduced relative to its free mass by the energy binding the muon to the muonic atom. This causes a decrease in the decay phase space for the bound muon, resulting in a smaller decay probability. (22) A  where AB = \mp{Z a)  2  bound  =  ^  _  A  B  ^ 5  (23)  > 0 is the binding energy; a is the fine-structure constant.  Relativistic time dilation due to the muon's momentum in the ls orbital further reduces the decay probability. The decay-electron energy spectrum arising from bound decay also differs from that associated with a free decay. Firstly, the orbital motion of the muon in the K-shell produces a Doppler-shift on the high-energy side of the spectrum extending it past the cut-off energy (52.8 MeV) for free decay, i.e. a "Doppler smearing" of the high-energy side occurs. (See figure 5.) Secondly, the nuclear coulomb field causes a shift in the peak of this spectrum; the peak moves progressively towards the low energy side as the Z value increases. These effects invariably influence the electron asymmetry from bound decay as well. Figure 6 shows the theoretical A - from the results of Gilinsky and e  Mathews [20]. It is noted that the correction due to the bound-decay-effect is relatively small; in the case of Ca, 40  this correction amounts to ~ 20%. The table I presents the  14  o  o  o  o  o  o  o  o  °  o  (x)9pniTTdrav  Figure 5: Theoretical decay-electron spectra for \i~ in lead and iron. In this figure, x = E7 /52.S. J  15  e  16  Table I: Normalised decay rates /N -(E)dE, integrated asymmetry coefficient / A -(E)dE and their ratios. The results quoted in this table are those of R. Watanabe et al. [23]. e  e  ie  Nucleus  Free  JN -(E)dE JA -(E)dE J A -(E)dE/ f N -(E)dE  1 -0.333 -0.333  e  e  e  e  0  0.989 -0.328 -0.332  40  Ca  0.981 -0.325 -0.331  90  208p  Zr  0.936 -0.306 -0.327  b  0.847 -0.262 -0.310  theoretical results for the decay rates and integrated asymmetry coefficients for several nuclei; calculated very recently by Morita's group [23].  I I . 5 N u c l e a r M u o n C a p t u r e a n dt h e R M C II.5.1  N u c l e a r M u o n C a p t u r e a n d its T h e o r e t i c a l T r e a t m e n t  The process of nuclear capture of a muon via the weak interaction (i.e. the OMC process) is represented by (A, Z)+ (*--> (A, Z-l)*  + !/„  (24)  where A is the mass number and Z is the atomic number of the parent nucleus. The daughter nucleus so formed is normally in an excited state (usually a giant dipole resonance state) which de-excites by emitting neutrons; this results in the major source of neutron background which is encountered in RMC experiments. In RMC, the final state as expressed in equation 24 contains a photon and the reaction is analogue of the elementary process a~ +p —> n + + OMC: ARMCIAOMC  7. RMC is far less likely than  ~ 2 x 10 , and so experimentally more difficult to study. In -5  this work we concentrate on RMC only. In nuclear muon capture experiments, some of the possible observables are 17  the total capture rate, polarisation, asymmetry and the energy distribution of the emitted nucleons. With RMC experiments, different aspects associated with the emitted photons are of interest. The first theoretical attempt at evaluating the total capture rate for nuclei was carried out by Wheeler [24]. He assumed that the capture rate was proportional to the strength of the muon wavefunction at the nucleus in the following sense:  E \m\  Ac«  2  (25)  Vprotons  For a hydrogenic wavefunction, ^(O)! ~ Z . 2  z  In heavy nuclei, a muon in a  ls orbital will spend a good deal of time inside the nucleus. Gauss's theorem for electrostatics shows that the electrostatic field felt by the fi~ depends only on the electric charge confined within its orbit. Consequently, Z is to be replaced by some effective charge (Z ff). e  For light nuclei Z yy ftiZ. Hence the capture rate is e  A <xZ .Z* e  eff  = Zt  ff  fJ  (26)  The muon capture rate is further reduced by the Pauli exclusion principle and nuclear recoil effects. This simple analysis illustrates how the capture rate is strongly influenced by the nuclear charge, and the reason why heavy nuclei are so much favoured by experimentalists. Following is a discussion of some of the approximate theoretical methods which allows one to obtain estimates for the RMC observables.  There are, in general,  two alternative methods of evaluating the elementary amplitudes for nuclear muon capture: The "impulse approximation" and the "elementary particle" approach. 18  • The Impulse Approximation (IA): The impulse approximation employs the methods of classical nuclear physics using nuclear wavefunctions, sum rules etc. Here an effective Hamiltonian is constructed from a simple sum over the quasi-free nucleons. The nuclear transition amplitude between the initial state (\ffii)) and final state (|V>/))  is given by  T . = <lMffe//llM  (27)  /f  where the H f j is the nuclear transition operator describing the muon capture proe  cess and accounts for nuclear structure effects. In the IA approach, equation 27 becomes Tfi =  Uf  A  (28)  V>.) t=i  where the transition matrix has been obtained from a sum over all quasi-free nucleons, ignoring multiple-scattering effects. The (non-relativistic) nuclear wavefunction ipi and tfrj are obtained by solving the Schrodinger equation for specified models: H i/> = Eil>  (29)  N  where HN is the model nuclear Hamiltonian. • The Elementary Particle Approach (EPA): In this procedure, one writes a general expression for the matrix element assuming that the nucleus is a structureless object characterised by such properties as parity, spin, and isospin. Consequently, the underlying form factors now relate to the nucleus as a whole.  In fact one of the major difficulties of EPA is the  construction of these form factors which display a strong c7 -dependence (q is the 2  4-momentum transfer). 19  Even though EPA is a seemingly very attractive and desirable approach, it has been pointed out [25,26] that it lacks serious justification for two step processes (such as RMC) in its present form. The technique is not wrong but simply insufficient. This is primarily due to two reasons. Firstly, we have a poor knowledge of the g -dependence of the form factors. Secondly, the application of the Born approx2  imation in low-energy theorems leaves undetermined terms which are numerically important. As a result, the majority of the calculations pertaining to RMC have employed the IA which is free from the above difficulties.  II.5.2  R M C Within the Standard IA  There exist several different models for treating RMC within the framework of the standard IA. • Closure Models: These are the simplest and the most often used models in conjunction with the IA. Basically, the closure models deal with how the summation over the final nuclear states is done. Here, it is assumed that all of the existing nuclear final states are attainable, i.e.  IVvKVvl  =  lj  *  n e  maximum photon energy is then expressed  in term of a common mean excitation energy ( A E ) : E™  ax  = k  max  = m -(AE) ll  (30)  The pioneering work for this approach was carried out by Rood and Tolhoek [5] (RT henceforth), where they used harmonic-oscillator wavefunctions. Furthermore, their approach to evaluate the amplitude of the elementary process p~ + p —> n +  +7  was a "diagramatic one" for which they employed the Feynman diagrams shown in figure 7. Diagram (f) corresponds to the induced-pseudoscalar coupling constant, 9P-  20  (a)  (b)  (c)  Figure 7: The Feynman diagrams employed by Rood and Tolhoek [5] in their diagramatic approach to RMC amplitude evaluation. Diagram (f) corresponds to the induced pseudoscalar coupling constant, gp.  2 1  (d)  The problem with early calculations such as RT was that they predicted absolute RMC and OMC rates which were too high. This resulted in negative values for gp/gA when experimentally-obtained data were fitted to the predictions of these theories. (This is in direct conflict with the Goldberger-Treiman [7] prediction where gp/gA  ~  +7.)  An attempt to remedy this complication was undertaken by Rood, Yano and Yano [27] (RYY henceforth.)  Their calculation was essentially the same as that  of RT; the only difference lies in the way the muon propagator was treated in the nuclear coulomb field. RT used the free muon Green function, while RYY solved for the Green function in the field of an extended nucleus. In this way, RYY obtained an ~ 25% reduction in the rate of high-energy photons. This decrease was sufficient to produce positive gp/gA results. Fearing [28] approached the problem of calculating the photon spectrum and the total rate for RMC in C a by using the Giant Dipole Resonance (GDR) model. 4 0  In this case, the dipole parts of the nuclear matrix elements relevant to RMC were related to integrals over the experimental photo-absorption cross-sections in the region dominated by the GDR. The multipole and velocity parts of the nuclear matrix elements were once again evaluated in the closure approximation using harmonic oscillator wavefunctions.  The most notable effect of including the GDR was to  reduce the absolute RMC rate by ~40% and the relative rates of R M C / O M C by some ~20%. An improved version of this model, due to Sloboda and Fearing [29], included correction terms of the order (1/m ), where m is the nucleon mass. 2  All of the aforementioned models treat k  max  as a free parameter, and so present  some difficulties when experimental data (with limited statistics and moderate resolutions) are to be compared with the theoretical spectra. This is particularly true in the case of the RT model which has a sensitive to the maximum photon energy. 22  power dependence and so is very  • Phenomenological Models: Christillin et al. [30,31,32] have criticised closure models on purely theoretical grounds. They argue that the closure sums include states of nuclear excitation which are simply energetically forbidden. As the RMC photon energy increases, the allowed part of the excitation energy decreases. This explains why all of the closure approximations overestimate the rates. Instead, they proposed a phenomenological nuclear response function which divided the transition strength into a dipole and a quadrupole part. The dipole resonance was a Lorentzian of width 5 MeV at 20 MeV excitation energy and the quadrupole resonance at 35 MeV had a width of 12 MeV. The quadrupole strength was chosen to produce the correct OMC rate. In this way, energies were properly accounted for. With photon energies greater than 75 MeV, only the dipole term contributes while for lower energies both are present. • Microscopic Models: This is the last calculational approach which will be discussed in relation to the IA approach. Gmitro et al. [6] employed detailed wavefunctions for the initial and all important final nuclear states. In this way, they performed a detailed microscopic calculation summed over all the partial transitions. However, this technique also suffered from an overestimation of the rate.  II.5.3  Beyond the IA  Attempts have been made to improve upon the naive IA scheme (or at least estimate the corrections to it) which neglects interactions involving more than one nucleons. Such efforts could be classified into three categories: (i) an estimation of the meson exchange correction (MEC), (ii) the use of effective hadronic form factors determined in some "renormalised" model within the framework of the impulse 23  approximation and (iii) the already mentioned elementary particle approach. It is noted that (i) and (ii) are indeed interrelated. In connection with category (i), the work of Gmitro et al. [33] is mentioned; this is an example of the Modified Impulse Approximation (MIA) scheme. The motivation for this work was to remove the discrepancy which existed between the (overestimated) rate as predicted by their work based on the IA scheme [6] and that of the phenomenological model used by Christillin et al. [32]. They achieved this by implementing the constraints dictated by the electromagnetic current continuity equation.  II.5.4  The Physically Measurable Observable, cv  7  We now turn our attention to the interesting observable of RMC, namely the Physical (or energy-averaged) photon asymmetry, cv . The energy-dependent RMC pho7  ton angular distribution is given by W (fc, 6)=  1 + r (Jb) cos0  7  (31)  7  where k = JE/ is the photon energy, 8 is the angle between the direction of emission 7  of the RMC photon and the direction of the spin-quantisation axis of fj,~, and T (A;) 7  is the energy-dependent photon asymmetry (coefficient.) T^(k) is defined in terms of the RMC rate (A c)  as follows  RM  r  ^  { RMc/ k)  - (dA /dk)  d  dA  =  ( ^RMc/ ) d  where A^  MC  = A c(k,0 RM  ^  RMC  dk  +  = A c(k,8  = 0) and A  RM  RMC  ( ARMc/ k) d  d  = TT).  Due to the low rate associated with RMC, an energy-dependent measurement of the photon asymmetry is experimentally not feasible. Furthermore, below 57 MeV, the experimental RMC energy spectrum is overpowered by bremsstrahlung 24  Table II: The energy-averaged photon asymmetry as calculated from several different models.  OJ.y  Reference  0.75 0.78 0.76 0.79 0.77  RT [5] RYY [27] Sloboda and Fearing [29] Christillin et al. [32] Gmitro et al. [6]  photons which thus restricts the range over which the RMC spectrum can be accurately measured. Consequently, OJ is defined over the experimentally accessible 7  region of k > 57 MeV as  _ Jfc>57 F-yJk) ARMC dk  (33)  /fc>57 A-RMC dk  7  For 100% polarised yT before capture (experimentally not realisable), the angular distribution of the emitted RMC photons in terms of a is then give by: 7  W (0)  = 1 + <* cosO  7  7  (34)  It is noted that o; — +1 exactly, for a pure V—A interaction. 7  Table II shows the values for a evaluated based on several different models. 7  It is seen that a is relatively insensitive to the underlying nuclear model used in 1  the calculations. Figure 8 shows ct as a function of gp/gA for both the Christillin 7  et al. [32] and Gmitro et al. [6] models. Since the average residual polarisation of H~ before nuclear capture is much reduced ((P -) ~ 10%), equation 34 has to be M  25  CN  CN  o q  | \  o  o  LO  LO CN  o  CD  o  d  q  L  Figure 8: The energy-averaged photon asymmetry as a function of gp/gA for the Christillin et al. [32] and the Gmitro et al. [6] models. 26  appropriately modified. The photon angular yield for RMC then becomes: y (0) 7  =  1 + ((Pp-) oi~f) cos0  =  1 + Ay cosd  (35)  The parameter Ay, which is referred to as the "observed photon asymmetry," determines the strength (or magnitude) of the modulation measured in the RMC-photon time-spectrum. As mentioned previously, this asymmetry occurs because the weak force does not conserve parity.  II.5.5  Possible Renormalisation of gp in Nuclear Medium  Before discussing the possible renormalisability of gp in the nuclear medium, it is useful to consider briefly the constraints which exist on the coupling constants in the process of fi~-capture. The Conserved Vector Current (CVC) hypothesis [34], d\V\ = 0, predicts that (36)  9s(q ) = 0 2  The Isotriplet Vector Current (IVC) hypothesis [34] assumes that V\ and the isovector part of the electromagnetic current to form an isospin 1 triplet, thereby allowing gv(q ) and ghi(q ) to be expressed in terms of the electromagnetic form 2  2  factors [35]: G (q ) 2  D  9v(q ) 2  (1 + <2 /4m ) 2  2  1+  4m  G {q ) (1 + q /4m )  (37)  2  2  9M(Q ) 2  =  D  2  Am  2  2  (38)  where -2  G (q ) 2  D  =  1+  0.71(GeV ) J 2  (39)  fi and fj, are the anomalous magnetic moments of the proton and neutron. At p  n  27  q  2  = 0  and q « 0 . 9 m 2  2  (which corresponds to the case of / i -capture by a free  proton)  flfv(O) = 1  9M(0) = n - f i P  n  (40)  = 3.706  (41)  and 0 ( O . 9 m J ) = 0.978 V  c 7 ( 0 . 9 m ) = 0.969( fi -y -l) 2  M  p  n  (42)  (43)  The gy, gA-, gp, and gM interaction terms are said to give rise to "first-class currents." The interaction terms associated with gj and g$ have been labelled as "second-class currents" [36]. The conservation of G-parity is generally hypothesised; it implies that  g (q ) 2  T  = 0.  (44)  gx = 0 = gs is the statement of the absense of second-class currents [36]. Finally, Partially Conserved Axial vector Current (PCAC) places restriction on gA and gp. P C A C states that  dA x  x  = fvmlfa  (45)  where f„ is the pion decay constant and fa is the pion field. This shows that the conservation of the axial current leads to zero probability for 7r-deay. Alternatively, it could be said that the axial vector current is conseved in the limit of zero pion mass (i.e. m —> 0.) PCAC in conjunction with the Goldberger-Trieman relation [7] n  28  relates un-renormalised (i.e. "free") gp and gA according to the relation 9A(q )  gp(q )  2  2  with the substitution of q  2  (46)  -0.9m in the above equation the Goldberger-Treiman 2  value is obtained: g  P  «  7g  A  (47)  Despite possible nuclear-structure-sensitivity of the meson exchange corrections, attempts have been made to produce renormalised theories by absorbing these corrctions, and replacing the free coupling constants by "effective" or "renormalised" ones, while keeping the structure of the impulse approximation for describing lepton-nuclear interactions. The two sources responsible for the possible renormalisation effect of gA and gp are (i) scattering of virtual pions in nuclear matter and the consequent reduced range for the pions, (ii)"Lorentz-Lorenz effect" causing a decrease in the strength of the pion source. The range of the pion-propagator (inducing the pseudoscalar interaction) is reduced in nuclear medium due to its interaction with the surrounding nucleons. In terms of the nuclear polarisability (cv), the pion's effective mass is expressed as  where in analogy with electrostatics, the strength of the pionfieldhas been adjusted by 1/(1 +a). The nucleons (which are the sources of pionfields)reside in "correlation holes" produced by the strong short-range-repulsive nuclear force. An effect (similar to that of the Lorentz-Lorenz in electrostatics where the strength of an electric dipole outside a cavity is modified by the dielectric medium), produces a renormalisation effect in the nuclear medium by reducing the strength of the pion source. This 29  -  results in short-range correlation effects which are expressed in terms of the LorentzLorenz screening parameter £. A value of £ = 1 implies full effect while £ = 0 means a complete suppression. The combination of the above two effects results in a possible renormalisation of the coupling constants in the form [37]  ^ = »^(i+|f«) 9p  =  g  p  (  1  +  l  (  a  )  (  1  +  a  )  - 0 ^ - -  («) 2  ( ) 60  where g^p (= 9A,P) are the effective axial and pseudoscalar coupling constants and gA,p (= g^p) stand for the corresponding free coupling constants. The numerical value of £ is considered to be rather uncertain [38]. However, the ratio of gp/gA is independent of £, and so provides a measure for a.  With  Ericson's calculated value of cv = —0.75 for infinite nuclear medium [9], one expects (with £ = 1) substantial renormalisation for the coupling constants: — « 0.75  (51)  — w 0.3  (52)  9A  9P  30  C h a p t e r III D e s c r i p t i o n of t h e E x p e r i m e n t The experiment described in this thesis was a study of the gamma-ray asymmetry in Radiative Muon Capture (RMC) on C a . It was carried out using backward 40  muon beam from the M20A channel at TRIUMF.  III.l B e a m productionand the M u o n  Channel  TRIUMF is a cyclotron facility capable of simultaneous production of two (variable energy) primary proton beams. It has a cyclotron radio frequency (RF) of 23.055 MHz (corresponding to a time separation of 43.37 nsec between consecutive beam bursts.) This is achieved by accelerating H ions to a maximum of 520 MeV -  and then passing the H~ beam through stripper foil to produce the protons which are ultimately extracted into the primary beam lines. The muon channel (M20A) is fed by one such primary beam line known as BL1A, which is capable of delivering 500 MeV protons at a peak current of 140 //A. BL1A has two meson production targets: l A t l and lAt2, the latter serves several secondary beam lines including M20A. In a typical muon experiment in the M20A area, the proton beam is allowed to strike a pion production target placed at lAt2; thereby producing pions and their decay product, muons, which are then collected and transported to the experimental area by means of bending and quadrupole magnets . Typically, the production target would be a water cooled strip of beryllium, 10 cm thick in the beam direction and 5 mm x 15 mm in cross sectional area. (Beryllium is used for its relatively 31  high pion yield and a low electron contamination of the pion beam.)  Figure 9  shows the general layout of the M20 channel with a simplified version presented in figure 10. By appropriately adjusting the bending and quadrupole magnets and the other available beam line control elements, namely the "jaws" and the "slits," it is possible to obtain a pion or muon beam with certain characteristics— spot size, flux, and divergence. One such setting of these control elements constitutes what is known as a "tune." A straight-through tune is one in which the second bending magnet, B2, directs the particles which have been momentum selected by the first bending magnet, B l , into the experimental area. Such a tune favours a pion dominated beam. A decay tune provides preferential selection for the muons which have resulted from the decay of the pions between the bending magnets. This type of tune is predominantly made up of backward muons. Depending on their mode of production, /i 's are classified as "cloud," "sur+  face" and "conventional," while /i~'s are only producible in the cloud or conventional forms. The conventional mode is derived from decay-in-flight of positive or negative pions in the decay section of the channel, and is further subclassified into "forward" and "backward" muons. The muons which arise from decays along the direction of the propagation of the parent pion in the Centre of Mass frame, are the forward muons, and those ejected opposite to pion's momentum form the backward muons. The decay possesses an isotropic angular distribution when viewed in the Centre of Mass frame, while it is sharply forward peaked due to the Lorentz boost, when observed in the Laboratory frame. Both forward and backward muons are emitted parallel to the direction of the boost as seen in the Laboratory frame, with the backward muons having the lower energy. The reasons for favouring a backward muon beam in this experiment were manifold . The lower momentum associated with such a beam makes the preferential selection of the muons and the discrimination against contaminants such as pions 32  CO  w  Q -  Quadrupole  Magnet  M20A Experimental  B J  Area  B e n d i n g Magnet  - Jaws  VA -  Valve  SL -  Slit  M20B  ecra  BB -  Beam B l o c k e r  3  t—'  o  in CO  tr: th  rt> Cu  to o o tr  ft  —  r  line  IA  T2  n  _^ >  Proton  Bean  and electrons more effective, resulting in a "cleaner" beam than is possible with either of the forward or cloud muons; this purity is achieved however with a loss of intensity. The lower momentum further implies a more effective stopping of muons in the target.  III.2  Experimental Geometry, Technique and  Philosophy  This experiment evolved out of a series of earlier experiments [39] that aimed at exploring the same physics. In this section, its general philosophy and physical layout are described. The main features of the geometrical set up are shown in figure 11. The experiment was primarily designed for the observation and extraction of the photon asymmetry, and for this, the muon spin rotation technique (fiSTV) [40] was employed. Here, the preferential decay along the direction of helicity of the muon is exploited in the following manner. The muon beam enters the experimental area with its polarisation vector essentially longitudinal and is then allowed to strike a target situated in a uniform transverse magnetic field. The average polarisation for the ii -beam in M20A has been estimated to be (P + ) = 0.75 [39]; this value being +  M  a function of the focusing and collecting elements of the channel. For those muons stopped within the target, the field causes their "spins" to precess (in a horizontal plane) at a uniform rate. By observing the RMC photons in a stationary detector placed at a fixed angle to the direction of the incident muon beam, the photon asymmetry would then manifest itself as a periodic time component superimposed upon a normal exponential decay curve in the resulting time spectrum. In this experiment the target consisted of two geometrically identical slabs of natural calcium (97% Ca) protected by a thin coating of aluminum. Each slab 40  measured 10 cm x 10 cm x 1.9 cm, and the two slabs were placed end to end to form a longer target of integral size 20 cm x 10 cm x 1.9 cm. The reason for doing 35  Figure 11: The experimental geometry. 36  so will be elaborated upon in the section under the heading of "Target Design." The desired magnetic field was applied by means of a small C-magnet (HASIMAG) specifically designed and constructed for this experiment; it provided a field homogeneity of 3% across the target. Its square pole faces measured 25.4 cm x 25.4 cm and were separated by a uniform gap of 15.2 cm. These pole faces were completely covered with 0.32 cm thick sheets of lead, so that muons which missed the target would stop in lead rather than iron. This would then produce a much faster decaying background time component which was easier to handle. The magnet was positioned with the centre of its pole faces 3.8 cm away from the beam's focus, and had its stand firmly bolted to the floor. The target, sandwiched between two detectors, S2 and S3, was then placed between the magnet's pole faces such that its geometrical centre coincided with that of the gap. It had its cross sectional plane of 20 cm x 10 cm vertical and was situated at 30° to the direction of the incident beam; this angle was optimally determined and more would be said about its choice later. The target's orientation with respect to the incident beam is shown schematically in figure 13. The precession field had the nominal value of 0.041 T, resulting in a precession frequency of 5.5 MHz. Like the earlier experiments, it had been decided not to employ the stroboscopic method [41] of //SR in which the muon precession frequency is tuned to equal that of the cyclotron frequency. Since R M C has a small rate (Rfc>57 ~ 2 x 10 ), it was a prime consideration -5  for the design of this experiment to improve upon the existing photon collection efficiency of the set up. This was achieved by enlarging the photon collection solid angle of the apparatus. The increase was done in two ways. Firstly, the construction of the new C-magnet allowed the photon telescope to be brought much closer to the target, and hence to provide a larger viewing solid angle of the calcium target by the detector.  This effect was maximised by placing the photon telescope at 37  90° to the direction of the incident beam, and as close as geometrically possible to the magnet. Secondly, the photon telescope itself was made more compact, hence further increasing the solid angle. (This was brought about by the removal of a lucite Cerenkov counter; see section III.4.)  The detection task for this experiment consisted, on the one hand, of identifying the arrival time and the ultimate fate of a (muon) projectile particle in terms of its stopping or passage through the target, of selecting what was to be a relevant event (charged or neutral) arising from the absorption or decay of the muon , and recording the time and energy information for this event. The former job was carried out by six plastic scintillators S0-S5, and the latter was reserved for the more complex assembly of detectors known as the photon telescope; its detailed description is given in section III.3. In addition there were other scintillation counters which acted as vetoes, counters to reject undesired background events. A detailed listing of all counters is given in the following section.  This section is concluded by a brief mention of the task of energy calibration. In order to be able to obtain an estimate of the total energy associated with a particular event, it is necessary to have an energy calibration for each of the detectors. This is done by studying a known energy spectrum. The thin and active Nal converter (BARS) was calibrated by allowing a u. beam , of known energy, +  to be incident directly upon each element of this segmented detector. The large total energy counter (TINA) was calibrated using the known spectrum for the n~ p reaction. For this calibration, the Ca 40  target and its associated counters, S2 and  S3, were replaced by a high pressure (100 atmosphere maximum) H gas target; the 2  gas target is shown in figure 12, and its detailed description is given in [42]. 38  oluminum  NE 110 cup  GAS  Figure 12: The gas target.  III.3  T h e Detectors  In this experiment, two types of scintillation counters were used; plastic organic scintillators and Nal inorganic scintillators. The plastic scintillators were used simply to detect the passage of charged particles. The other detectors, based on thalliumactivated sodium iodide scintillation material, Nal(Tl), were employed for photon detection and and energy measurements. A total of 29 plastic scintillators of varying sizes, all viewed by RCA8575R phototubes, were used. A partial listing of these counters and their dimensions is given in table III. In addition, there was one additional counter with dimensions 38.1 cm x 38.1 cm x 0.64 cm with a central hole of diameter 15.2 cm, called the Hole counter (H). It was furnished on both sides with two geometrically identical waveshifters bars each measuring 3.2 cm x 38.1 cm x 0.64 cm which resulted in an overall 39  Table III: Plastic scintillation counters dimensions. Counter SO SI S2 S3 S4 S5 E VAB LST1 LST2 LST3  Size (cm ) 3  25.0 25.0 27.9 33.0 33.0 33.0  x x x x x x  25.0 25.0 12.7 12.7 29.5 29.5  x x x x x x  0.32 0.32 0.16 0.16 0.32 0.32  12.7 0 x 0.32 58.0 38.0 38.0 38.0  x x x x  48.0 38.0 38.0 38.0  x x x x  0.32 0.32 0.32 0.32  C1-C6 C7-C12 C13-C16  44.5 x 30.5 x 1.30 78.7 x 30.5 x 1.30 100.3 x 40.6 x 1.30  GTS  3.20 x 3.20 x 0.32  physical size of 44.5 cm x 38.1 cm x 0.64 cm for this detector. The beam/target counters, S0-S5, were used to detect the arrival of a muon and its stopping or passage through the target. The beam counters, SO and SI, were sufficiently large to completely cover the beam as it emerged from the lead collimator (placed in front of the snout at the end of the beam pipe), and the target counters, S2 and S3, had the calcium target sandwiched between them with its front face covered completely. The other two target-associated counters, S4 and S5, were slighty larger than the magnet pole faces which they covered. S4 and S5, in association with S2 and S3, were used to define a muon stop in the target. A signal in SO, SI and S2 with no signal in either of S3, S4 or S5 implied a stop, whereas a signal in any of S3, S4 or 40  S5 meant a passage through. S3, S4, and S5 were further used to discriminate against bremsstrahlung radiation due to charged particles in the target which could falsely replicate good neutral events originating from within the target. The photon telescope selected neutral events (including RMC photons) arising from the interaction of muons inside the target, thereby allowing the construction of an RMC photon time spectrum and the extraction of the photon asymmetry from it. The telescope had two main components: TINA, and the BARS; both Nal(Tl) detectors. TINA (TRIUMF Iodide of Natrium) is a cylindrical crystal of size 46 cm 0 x 51 cm, enclosed in an aluminum container which is viewed by seven 12.5 cm RCA S83006E phototubes. A useful reference on its use and performance is [43]. Because of its comparatively large geometrical volume, it absorbed most of the energy associated with a given event. The BARS consisted of seven geometrically identical and optically isolated blocks of Nal(Tl) of total assembled dimensions of 5.08 cmx 30.48 cmx 35.56 cm, with each individual block measuring at 5.08 cmx4.35 cmX 35.56 cm and viewed by 3.8 cm Hamamatsu R980 photomultiplier tubes at each end. Like TINA, it is also encapsuled in an aluminum container. It was used as our gamma converter, turning photons into electron-positron pairs. [44,45] serve as useful references on some performance features of scintillation assemblies such as the BARS. By requiring simultaneous energy deposition (E > 5MeV) in both TINA and the BARS, it was possible to discriminate against most of the neutron background arising from OMC in the target and from other sources. The other elements in the photon telescope were the Layered ScinTillator (LST), the Veto counter (VAB), and the hole counter. The LST consisted of a triplet of geometrically identical pieces of plastic scintillator (LST1, LST2, LST3) 41  optically isolated from each other and viewed by separate phototubes; its function was to detect the charged particles arising from photon-conversion in the BARS, thereby allowing an estimate of the number of leptons per conversion event (from a measurement of the total energy deposited) , and hence to provide further discrimination against neutron background. The VAB, placed in front of the BARS, vetoed charged particles; because of its large width, it had two phototubes for a more efficient collection of the scintillation light. TINA, BARS, and the LST were assembled inside a 12.7 cm thick iron shielding box called the "TINA carriage." It had a removable lid and a 30.5 cm 0 collimator in its front face. The arrangement was quite compact so that the neighbouring elements touched; this in order to maximise the solid angle. The hole counter was designed and placed outside of the shielding box in front of the 30.5 cm 0 collimator so as to identify charged particles which might create bremsstrahlung radiation in the collimator (giving rise to false neutral events.) The  photon telescope was equipped with cosmic-ray vetoing counters  (C1-C16); one layer of these detectors was mounted on the outside of the shielding box (10 counters plus the H), and another layer was placed on the inside (6 counters plus the LST) thereby completely covering TINA. These formed the so called "OUT" and "IN" counters respectively. A coincidence between the logical-sum of the IN-counters and OUT-counters meant the passage of a cosmic-ray through the apparatus. In addition to detecting RMC photons, the photon telescope also detected decay-electron events which determine the polarisation of the muons just before capture; this serves as the normalisation factor for the photon asymmetry. With the large rate of the muon decay (and the associated charged events) as compared to RMC, it was necessary to regulate the detection rate of the former. In order to give the telescope this desired capacity, it was furnished additionally with an 42  electron counter (E); a small circular counter placed in front of the hole counter which defined a reduced solid angle (and hence a reduced rate) for the collection of decay electron events. The gas target scintillator (GTS) was employed as a defining counter for the gas target and it was used only during the TINA calibration runs. When placed in contact with the front face of the gas target, it gave a more precise definition of a stop in the gas.  III.4  Design of the  Experiment  In this section, the design aspects for this experiment are discussed. Included also is a description of the background and the shielding designed to guard against it. The construction of the C-magnet and the resulting increased solid angle has been discussed earlier in the section III.2. The other component in the gain of the solid angle was brought about by the removal of a lucite Cerenkov counter from the photon telescope assembly. The Cerenkov counter had originally been included due to the uncertainty about the effectiveness of the LST in identifying the charged particles arising from photon-conversion in the BARS. A detailed analysis of the earlier runs allowed its removal without any consequent loss of photon-converted events in the photon telescope.  III.4.1  Target Design  In designing the target, there were two conditions in mind. Firstly, to have as large a fraction of the projectile particles stopping in the target as possible. Secondly, with the photon telescope viewing the back face of the target, it was desirable to have the centroid of the stopping distribution (along the thickness of the target) as close as possible to this face; in order to reduce the interaction of the decay electrons within the target before they emerged from the back face.  43  This would reduce  brernsstrahlung within the target and also would allow a better determination of the i i polarisation before capture (less Monte Carlo dependence.) -  Using C A L C T G T [46] (programmed by the author for optimisation studies of the beam stopping distribution and employed by Virtue [39] in his analysis of the TRIUMF's earlier RMC experiment preceding this work), it became evident that with our given set of experimental conditions ((p - ) = 76 MeV/c and 86 MeV/c, M  target inclination 20° <0<  60°, a magnetic field of 0.041 T, and of course the given  rectangular shape of the calcium target pieces), the mean penetration depth ranged from 0.7 cm to 1.6 cm, i.e. well within the thickness of one target slab. In the light of this new finding, it was therefore decided to depart from the earlier choice of a "thick" 10 cm xlO cmx3.8 cm target, and to employ a "thin-and-extended" target obtained by placing the slabs end to end resulting in a compound target of size 20 cmxlO cmxl.9 cm. Figure 13 shows the relevant geometry with the following notes making reference to it: • C : centroid of the stopping distribution. • O : geometrical centre of the compound target. • x : indicates the direction to the photon telescope. • z : indicates the direction of the incident fx~ beam. • d : the "emergence length," i.e. the distance to target's back face in the s  direction of TINA. Having decided on the physical size of the target, it was now necessary to decide on its orientation with respect to the incident beam. The optimisation problem here was to choose 9 such that it minimised the emergence length while at the same time maximised the % stop within the target. No one clear optimum solution existed. It 44  Figure 13: Target geometry.  45  Table IV: Target inclination optimisation results. (p -) = 76 MeV/c % stop in C a d (cm) M  ( °) 60 50 55 40 30 20  40  a  43 56 61 65 68 69  0.7 0.6 0.8 0.9 1.2 1.4  ( - ) = 86 MeV/c % stop in C a d (cm) P<1  40  s  2 8 15 26 53 67  0.7 0.4 0.45 0.5 0.6 1.1  was decided to settle for 6 = 30°, a choice which provided an acceptable level for both of the requirements. A short summary of the optimisation results is given in table IV. (It is noted in passing that the Monte Carlo % stop values of 68% and 53% are in good agreement with the experimentally observed values of 70% and 53% respectively.) It was also of interest to know the effect of the beam scintillators thickness (0.16 cm or 0.32 cm), the relative separation, and the distance from the target to the beam counters (SO and Si) on the % stop and the mean muon penetration depth. This study showed an insensitivity to all of the aforementioned factors. In order to achieve the best shielding and collimation, it was decided to place SO and Si adjacent to each other and as close as possible to the target.  III.4.2  Shielding and Background Considerations  A major design concern for this experiment was that of the shielding. The background problem had been so serious for the preceding experiment that the asymmetry was detectable only with a signal to noise ratio of \ to 1. Here, the background problem is discussed only briefly; a more detailed description is to be found in [39]. But first, a few words on why the background should be such a problem. 46  Muon capture is not prompt with its associated muon stop event within the target; the process of capture takes place with a mean life time of ~ 333 nsec (in 40  Ca.) In order to be able to detect a reasonable proportion of these muon capture  events, it is necessary to observe them within a sufficiently long time period after each muon stop. In this experiment, the detection system was enabled to accepted any such occurrence for up to about three life times in C a (~ 1 //sec) following a 4 0  stop. To make things even worse, RMC in C a possesses a small branching ratio of 40  Rfc>57  ~ 2 x 10~ , thereby allowing other less probable, moderate processes to rival 5  it. Hence, this combination of low rate and finite life time made the problem of the background crucial and its reduction vitally important. The backgrounds encountered in this experiment could be categorised into three main classes: Cosmic-rays, beam (or cyclotron)-related, and target-associated backgrounds. Of these, there were two that one could go some way towards guarding against, and the third (target-associated) for which no such measure was possible. The experience gained in the past during earlier "runs" with the identification and exact nature of these sources was put into full use in designing the shielding as effectively as possible within the given resources available. One of the sources we were able to shield better against was the cyclotron related background, (presumably arising from the interaction of the p's and 7r~'s within the material of the beam pipes in BL1A and M20A, and 7r~'s and neutrons directly from the M20B2 bending magnet) having a structure with the characteristic cyclotron period of 43.37 nsec. The apparatus was partially protected from the above mentioned background by building a 183 cm wide and 213 cm high concrete/iron wall right up against the TINA carriage on the side of TINA closest to the BL1A, and a collimator section extending right out to the edge of the precession magnet coil with just sufficient space to insert the beam counters SO and Si. Furthermore, all the open space 47  remaining between the wall, the beam pipe, and the collimator was filled in by small lead bricks from the beam level up to a height of about 200 cm. This is schematically shown in figure 14. Additionally, lead bags were deposited wherever possible in the open spaces on the last three quadrupole of the M20A beam pipe for extra shielding. The wall was composed of standard 60 cm x 91 cm x 182 cm iron or concrete blocks, it was concrete up to the height of 60 cm, then iron from 60 cm to 210 cm; having its maximum thickness of 122 cm at the level of TINA crystal to provide the best shielding. The collimator served the dual purpose of giving a "good" beam definition and providing further shielding against background. It was constructed from small lead bricks and started from inside a U-shaped iron yoke (30 cm thick and 32 cm long in the beam direction) placed immediately in front of the beam snout; see figure 14. The collimator opening had a uniform rectangular vertical cross-section of height 10.3 cm, and a horizontal cross-section tapered from 15.2 cm to 10.2 cm downstream in the beam direction with a total length of 41 cm. This opening was formed in such a way as to envelope the beam at its Full Width \ Maximum (FWjM.) Immediately following this main section of the collimator, there was an adjoining uniform square opening of cross-section 14 cmx 14 cm and of length 8.9 cm (along the beam), included to avoid end scattering from the collimator. Now, the problem of the cosmic-ray background is discussed. A cosmic-ray arrival and a muon stop event bear no relation to each other timewise, this randomness results in a featureless and flat background time component. Cosmic-rays having a "cos#-distribution" (8 measured from the vertical), are most intense in the vertical (downwards). In this case, the shielding took the form of a concrete roof formed by two blocks of dimensions 122cm x 183cm x 549cm which completely covered the whole of the set up. Directly over TINA, the shielding had a total thickness of 244 cm. 48  Figure 14: Detailed shielding plan in the vicinity of M20A beamline. Scintillators H, E, S2, S3, S4, and S5 have been omitted for clarity. 49  A good deal of effort was invested in the design and the construction of the shielding for this experiment and we were accordingly rewarded by the improved signal-to-noise ratio of 7 to 1. The last topic in the discussion of background is that of the target-related background which resulted from undesirable interactions within and around the target. It was not possible to shield against this source, and one had to rely on the so called "software cuts" for its reduction. (See Chapter IV which deals with the analysis of the data.) For a muon which stops within the C a target, three possible alternatives 4 0  exist. It could decay in orbit (BR~ 15%) giving rise to an electron which could then bremsstrahlung in the target, or it could get captured in a C a nucleus (BR~ 85%) 4 0  resulting in the ejection of a neutron, andfinallyit could get absorbed "radiatively" in the nucleus  (Rk>57  ~ 2 x 10 ). -5  The bremsstrahlung component in the background becomes negligible for energies > 57 MeV; it is possible to further discriminate against it by studying the time spectra for S3, S4, and S5 and applying an appropriate software cut. The new choice of the thin-extended target was also aimed at reducing this source of background by shortening the effective path length of the decay electrons exiting the target. The Nal crystal is a good absorber of neutrons and because of this, the neutron background in this experiment which arises mainly (and inevitably) from OMC in the target was quite considerable. For its reduction we relied on the neutron discrimination ability of the photon telescope. The removal of neutrons was quite effectively implemented in the form of a hardware/software cut which required simultaneous energy deposition above a certain lower limit in both TINA and the BARS. Muons coming to rest in the region around the target, resulted in a background 50  with two distinct half lives, one short and the other long. The short lived component was due to the stops inside the lead plates on the pole faces (rpf, ~ 75 nsec [18]), whereas the long lived component resulted from stops in the carbon material of the scintillators and wrapping (r ~ 2.03 iisec [18].) These two half lives differ c  markedly from that in C a , and a several life time component fit to the neutral 40  time spectrum allowed an estimation of the total number of stops outside of the target. 7r 's interact promptly within matter upon coming to rest, and so it is easy -  to discriminate against most of the pion contamination in the beam by rejecting those events that are prompt with a particle crossing the beam counters SO and Si. The background due to 7r 's which miss SO and Si could not be dealt with in _  such a manner, and in fact one reason for having such a restrictive collimator in this experiment was to reduce, as far as possible, the number of beam particles that missed SO and Si and/or the target.  III.5  Electronicsand Data Acquisition  Before describing in detail the electronics set up and the data acquisiton system, it is useful to explain the concept of an "event" as used in relation to the electronics. Three subclasses of events were recorded: NEUtral (NEU), CHarGed (CHG), and PuLSer (PLS) events. The NEU event was intended for classifying those processes which were recognised as being electrically neutral at their point of creation inside the target. It aimed at the (RMC and bremsstrahlung) photons and the unavoidable and the undesirable (OMC) neutrons. The CHG event referenced charged decay electrons (or positrons) arising from muon decay in orbit. CHG and NEU events were further logically combined to produce the more 51  general GooD eVent (GDV). The PLS event was obtained by using a beam pulser to sample events at random (following the arrival of a beam particle) thereby allowing the determination of the pedestals on our ADC's (Analog to Digital Converter) (i.e. providing information about the noise level on the counters and the thresholds of the ADC's.) The PLS event is also a NULL event because it did not necessarily occur because of an actually relevant physical particle entering the gamma detector. The logical addition of GDV event and PLS event produced the so called "trigger event"which controlled the triggering of the electronics. The detectors experienced rates of order of 400K (for //stop) and had resolving time down to a few nsec. The signals from the counters were processed by the standard NIM modules (mainly LeCroy) and the relevant data was then passed via a C A M A C (Computer Automated Measurement and Control) crate to a PDP 11/34 computer which ultimately recorded this data onto magnetic tape. The two main programmes operating on the PDP 11/34 were DA (Data Acquisition) and MULTI. The former handled the reading of the data and its recording onto magnetic tape, while the latter allowed online analysis and inspection of the data (without affecting data taking) as the experiment was progressing. The computer system gave data collection and recording the highest priority over all other computer processes. The linear PhotoMultiplier Tube (PMT) signal from each detector was employed to provide us with both time and energy information about the passage of a particle through the detector. By feeding the linear signal directly into either a peak-sensing or charge-integrating ADC, we were able to measure the amount of energy deposited in the counter by the particle. Time information was obtained by sending the linear signal into a discriminator (a module which produces a logical signal of desired time width from any linear 52  signal above an adjustable threshold) to produce a logical signal which was then fed into a T D C (Time to Digital Converter.) The times so recorded by the TDC's were in fact time-differences. In this experiment, the (trigger) event set the timing, i.e. it was the zero of time and all other times were measured relative to it. (The event time itself was in turn determined by the BARS timing.) The TDC's are only capable of recording positive time differences, hence all of the signals that preceded the event signal had to be delayed using delay cables. For the plastic scintillators with their very fast response time (pulse rise time ~ 15 nsec), D's (leading edge Discriminator) were employed. However the slow response time of our Nal detectors (TINA typical pulse rise time ~ 80 nsec) necessitated the use of CFD's (Constant Fraction Discriminator), in which the timing of the output pulse is relatively independent of the size of the input pulse. There were seven signals from TINA and fourteen signals from the BARS, and given the pulse heights and rise times, it was necessary to have special provisions for processing of these multiple signals. TINA's individual signals were passively split, then one signal from each phototube was RC integrated and amplified 10 fold and finally passed onto a peaksensing ADC. The other signals were summed in a summing ampifier (with a gain of unity) and clipped at 288 nsec by reflection in a delay cable. This clipped signal was then sent to various peak-sensing and charge-integrating ADC's and a CFD. The output of this CFD then gave the TINA timing. In addition, the seven TINA dynode signals were passed through a switch box, then preamplified (in the experimental area) and subsequently amplified by a factor of 10 in an integrating and shaping amplifier. This hardware summed and amplified signal was then sent to a high-speed, high-resolution, ultra-linear buffered ADC (a LeCroy 3512 ADC) operating in the peak detect mode. After being amplified 100 fold (in the experimental area), theBARS signals 53  were passively split and one signal from each phototube was summed and the sumBARS signal was clipped at 260 nsec. The clipped signal was passed through a CFD to provide us with the BARS timing signal. Additionally, the individual BARS signals were sent to both peak-sensing ADC's, and TDC's to provide energy and time information about each of the bars for every event. In the previous experiments, the BARS and TINA signals were summed and passed (after setting a minimum energy threshold of ~ 25 MeV) to a CFD, the output signal from which was used in the NEU coincidence. This was done in order to reduce the amount of the data collected to a manageable level by cutting out the unwanted and prolific low energy events. For this experiment, we experienced serious timing problem associated with this "hardware cut" on the total energy and had to resort to employing a "software-tape-cut" on the energy instead. This was achieved by modifying the DA in such a way that, only events with energy greater than ~ 25 MeV were recorded onto tape. Figure 15 shows a simplified version of the electronics employed in processing of the BARS and the TINA signals, where for the sake of simplicity, the function of an identical group of modules has been represented schematically in terms of the function of only one of the members. The logical signals from the various discriminators in the circuit were used in the hardware event definition and formation. Here " . " denotes the logical AND operation (also referred to as coincidence formation), " + " is the logical OR operation, and "(...)" stands for the logical negation of "(...)". In connection with event formation it is worthwhile to briefly discuss the function of a "veto" and an "inhibit." They both are implemented by logical signals and have similar function. A veto is used to disable part of the electronics (normally a coincidence unit) for a specified length of time (usually of the order of a few tens of nsec), while an inhibit impairs the whole of the system for a predetermined span of time (in this experiment 54  to TINA P. S.  rrrn  TO N E U T R A L EVENT - ^ T D C  )  >TDC  I /X s  )  f INA^ITE-HP  1  ADC GATE  =-  |TINA n >  n r r n  ^ADF) • X |  •^Scoler)  pnrn  TO  2  —SID  NEUTRAL/  CHARGEDCOINICIDFNf  1  BARS  BARnL>-  BARS PILE-UP ADC GATE  IEVT  t r ^ L ^ ^ - i A D D  I  1  ADC  )  ->Scoler)  BARnRJJ  Figure 15: Electronics for processing the TINA and the BARS signals. 55  it varied from 300 nsec to 750 psec depending on the source of the inhibit.) So veto has the local task that inhibit plays globally. For brevity, the following abbreviations are henceforth used. NEU, CHG, GDV, and PLS are to stand for NEU event, etc. NEUV, CHGV are to mean NEU Veto and CHG Veto, respectively. In forming CHG and PLS, two separate and independent pulsers were used, namely the charged pulser and the beam pulser; the function of the former was to further reduce (in addition to the E counter) the charged event rate collection. The coincidence (50 • 51 • S2) signalled the arrival of a muon inside the target and [50 • 51 • 52 • (S3 + 54 + 55)] told us that the muon had in fact stopped within the target, i.e. a pstop. The following vetoes were used = 50 + 51 + H + VA + VB  (53)  = SO + 51 + H + charged pulser  (54)  NEUV  CHGV  where VA and VB refer to the individual phototubes of the veto counter VAB. The "elementary" events were NEU  CHG  = BARS GDV  PLS  = TINA  • BARS  -S3-E-  • NEUV  (55)  (VA + VB) • CHGV  (56)  = (NEU + CHG) • inhibit  = 50 • 51 • 52 • beam pulser • inhibit  (57) (58)  The "EVenT" (EVT) which triggered the CAMAC modules was compositely defined as EVT  = GDV + PLS  56  (59)  Figure 16: The trigger logic. 57  this being the most general form of an event defined in the experiment. The corresponding trigger logic is shown in figure 16. The decay of a captured muon was observed within a 2 jusec long time window generated by a Pile Up Gate (PUG) which was fed by GDV. The delays in the electronics were arranged so that the "prompt event" occured roughly half way into this time-window, giving us about three-life-time in C a time for recording a 4 0  possible decay. (The prompt was actually at 985 nsec, resulting in 985 nsec in the foreground and 1015 nsec in the early time background region.) During the very early stages of the experiment, the GDV formation included an additional component in the form of a 1.1 / i s e c long and extending gate originating from a //stop and applied via a PUG. With this set up, only a small portion of the early time window was available (~180 nsec.) The elimination of this gate from the coincidence defining the GDV provided us with a fully extended early time, offering a better estimate for the time-independent background. The muons arrived randomly at the target with their arrival-time distribution following Poisson statistics. If within the 2/xsec time window that we were awaiting a possible decay, a second (or third, etc.) muon arrived, we had no choice but to discard that particular event from the time spectrum in our data analysis, as there was no way of knowing which muon was responsible for the observed decay. We note in passing that the above event is suitable and in fact can be included in the so called "rate measurement." The module which enabled us to identify this muon pile up was the pulse separator; it is capable of "splitting up" a train of pulses into a series of separate pulses allowing relative time determination between consecutive pulses with a resolving time of 6 nsec. For more information on the different forms of pile up encountered in this experiment and the remedial actions taken to safeguard against them, the interested reader is referred to [39]. Figure 17 shows the pile up related part of the electronics. 58  Another item of interest in the electronics was the bit-register C212. It was built into the logic so that for a given type of event a unique bit was set, thereby allowing us, amongst other things, to identify the nature of E V T (i.e. whether NEU, CHG, or PLS.) It was "strobed" from E V T by a short and a long (~ 2/isec) strobe; the long strobe was intended for recording pile-up-related occurrences such as a second muon arrival or cosmic-ray event, etc. Promptly following an E V T , a long inhibit (750/isec) was established to provide sufficient time for the computer to respond, and while the C A M A C crate was being read, a second inhibit was actuated by the crate's NIM drivers allowing the computer uninterrupted access to the NIM modules. Once the computer was finished reading the crate, the NIM drivers released the inhibit and opened the electronics for the next event. In addition to the above two "data reading" inhibits, there was the additional "data taking" inhibit (300 nsec long) arising as a consequence of the signal (50 +51)-(54 + 55); this inhibit safeguarded against the background resulting from a muon stop in the lead plates on the pole faces. Each E V T was characterised by 104 numbers which were recorded onto a magnetic tape during the data acquisition; the numbers being the digitised values of the data words associated with the ADC's, the T D C ' s , and other modules (such as the C212.) In the MULTI's terminology, these were the "typel" events. MULTI recognises a second type of event known as the "type2" event which consists of scaler readings. Scalers were used to count the number of occurrences for items of interest (such as the number of muon stops, etc.)  They counted  continuously and were written onto the tape either when the register overflowed (a type2 event) or when a given run was ended.  59  •j. STOP  GG  1NHBIT  PUG  PUG  B  B  D  D  PUG  B  D  PUG  B  D  Figure 17: Schematics of the pile-up detection electronics. 60  ->TDC)  Chapter IV Data Analysis The R M C asymmetry measurements were carried out with a backward a~ beam at two different average beam momenta: 76 MeV/c and 86 MeV/c. The calibration data for TINA were obtained with a 100 MeV/c 7r~ beam, while the BARS was calibrated in an 86 MeV/c backward ju beam. The entire data set was recorded +  onto 65 magnetic tapes at a density of 6250 BPI. In this chapter the technique employed for extracting the desired photon asymmetry from this data is discussed.  I V .1  Introduction  With the application of optimised software cuts to the MULTI-written raw neutral data, a "clean" neutral time spectrum was constructed. The software cuts were energy and/or time constraints imposed on the data in terms of FORTRAN written routines (DPLOT's). They were implemented by the off-line analysis programme MOLLI [47] capable of reading and handling the MULTI-formatted data. These cuts were expressed as logical variables and could be categorised into two distinct classes: "event-selection logicals" and "cut logicals." The event-selection logicals in connection with a bit register (C212) allowed the identification of a particular event type (NEU, CHG, or PLS); with this type of logical, the truth value TRUE caused the event to be selected. On the other hand, the cut logicals were used to remove (or "ctti out") an event of any type when the truth value of the logical was set to TRUE. 61  The observed photon asymmetry (A ) was determined by fitting a (iSR func7  tion of the form N(t) — N e~r (1 + A cos(u>t + <f>)) + B 0  (60)  7  to the neutral time spectrum,where: • N(t) : # of events per unit time at time t. • No : # of events per unit time at time t=0. • r : decay life time of fi~ in C a . (~ 333.0 nsec) 40  • u> : angular precession frequency. • (j> : initial phase angle. (+90° for a.- in this experiment.) • B : background term. A  x "^ ^ ! ^ ) was the chosen technique for parameter determination, 2  1  111  33  011  and this was carried out with the use of the nonlinear functional-minimisation programme MINUIT [48]. The asymmetrical errors quoted for the estimated parameters are those provided by the MINOS subroutine of MINUIT which gives 1<T errors (68% confidence limit.) The quantity of interest is the physical photon asymmetry (a ). It is obtained 7  by normalising the observed photon asymmetry to the average u,~ polarisation before capture, i.e. the residual //"polarisation ( (P -) ): M  (61) The chosen value for this normalisation factor ( (P^- ) = O.llliO.010 ) is the value which was determined in a similar experiment at M20A which preceded this work [39]. 62  The first step in the analysis of the data consisted of removing the low energy neutral events and the neutron background. This comes under the general heading of "skimming." The process of skimming was performed with the help of skimming routines which are relatively standard at TRIUMF. These routines allowed one to pick out the events that passed the skimming requirements and write them onto a disk file and later onto another magnetic tape.  This initial skimming produced five 1  st  generation skimmed magnetic tapes at 6250 BPI. Skimming automatically removed all of the type2 events and so the scalers readings had to be printed out separately throughtout the skimming sessions. The neutron removal condition consisted of requiring > 5 MeV energy deposition simultaneously in TINA and the BARS with a total energy in both of not less than 25 MeV. Low energy neutral events were removed by requiring >45 MeV total energy deposition in the TINA and the BARS. This 1  st  generation skimmed neutral data set was subjected to one further  skimming operation requiring the so called "Layered Scintillator cut" condition; a detailed discussion of this cut is delayed until section IV.4. This reduced the data sample to one 2  nd  generation skimmed tape at 6250 BPI. The software cut  optimisation studies and the final fits were carried out on the clean neutral data set included on this one tape.  IV.2  Choice of the Fitting Function  The most general function describing the observed neutral time spectrum for this experiment takes the form  N  oi  e  r  ' (1 + Ayi e~  63  rit  cos (ut + if>))  + B  (62)  where the index i is to run over all of the possible sources radiating the RMC photons. t_ The exponential decay term, e >, arises because of the finite life time for T  the muon capture process. The periodicity inherent in the neutral time spectrum which is associated with the precession of the ^u's spin, is expressed by cos(o;t+<^). Any possible non-homogeneity in the precession magnetic field would make itself felt through a (time) relaxation of the signal and is accounted for by the relaxation term, e  -r,t  .  The main source of the RMC photons was the calcium target. There were two other minor sources which one had to worry about: the lead and the carbon components arising from p~ absorption in the lead sheath and the scintillators or their wrapping, respectively. From two-component fits to the neutral time spectrum, it was concluded that, within the existing error bounds, the above two minor sources could be omitted from the fits without affecting the estimated value of the observed asymmetry for C a . ( ^  ~ I O and  40  ~ IO" . ) Furthermore, no relaxation  -6  effects were observed (rc  a  3  ~ 10 ); a consequence of this is that the value of the -5  observed asymmetry is independent of the zero of time.  Hence, with these prevailing conditions, the fitting function becomes a single time component reduction of equation 62:  I V .3  N(t) =  N (t)  =  N e~r(l  Ca  0  + AyCos(u>t + </>)) +B  (63)  Calibrations  This section deals with the time calibration of the TDC's and the energy calibration of TINA, BARS, and the LST. The zero of energy for every ADC was identified with its associated "pedestal." 64  The value of the pedestal was provided by the centroid of the pulser distribution for the given ADC. The 3512-ADC used for the TINA hardware sum was an exception however. For this particular ADC, the value of the pedestal was determined by extrapolating the response curve of the ADC to a precision pulser down to zero pulse height.  IV.3.1  T i m e Calibrations  There were 4 TDC's employed in this experiment.  However, only 2 needed an  explicit time calibration. TDC3 (2 //sec full range) provided the Time of Flight (TOF) from the target to the BARS, while TDC4 (200 nsec full range) gave the //stop timing. A difference of the appropriate words from these two TDC's when histogrammed for the neutral events, resulted in the neutral time spectrum. TDC4 was easily calibrated by means of one of its words which recorded the Capacitive Probe (CP) timing. The CP is a beam monitor which produces a signal for each passage of a proton beam burst through it. The CP signal was chosen because it is a more reliable source of timing than the Radio Frequency (RF) signal which could shift in time due to the phase variation of the protons with respect to the cyclotron RF. A simple self-vetoing logic circuit in conjunction with the CP signal produced a double peaked spectrum for CP within the full range of the TDC4 with the peaks separated by the characteristic period of cyclotron at 43.37 nsec. In this way the calibration constant for TDC4 was determined to be 0.1000+0.0004 nsec channeljj:'  No such a facility existed for TDC3. Instead, it was calibrated in terms of the 43.37 nsec structured background which was obtained by selecting high energy events (> 80 MeV) that left no energy in the LST. The time calibration constant so determined was 1.01 It0.003 "^/#c/ia  65  IV.3.2  Energy Calibrations  Detectors such as the TINA and the BARS with multiple elements are operated with equal (or nearly so) gains for all of their photomultiplier tubes. The process of adjusting the high voltages on these different tubes in order to attain gain-equality is known as the "coarse or hardware balance." The off-line handling of the variation in the gains associated with different photomultiplier tubes is known as the "software balance." It is implemented by incorporating the so called "gain factors." The high voltage settings for the TINA tubes were those determined in a recent balance done by the 7T —• ev group." On the other hand, the BARS were U  explicitly balanced for this experiment in an 86 MeV/c backward p beam. +  For a given ADC, the calibration amounted to determining an energy calibration constant, E = K ( VADC e  - pds )  (64)  where: • E : value of the energy as recorded by the ADC. • VADC : digitised value produced by the ADC. • pds : the pedestal on the ADC. TINA was calibrated by stopping negative pions in hydrogen.  This was  achieved by positioning the gas target (filled with H gas at a pressure of 1100 psi) 2  in a 100 MeV/c ir~ beam. The energy spectrum resulting from the ir~p reaction at rest consists of a "7r°-box" and a "129.4 MeV-peak." See figure 18. The finite resolution of TINA, when convoluted with this ideal spectrum, produces the observed TINA energy spectrum. Figure 19 shows such a typical 66  experimentally-observed TINA energy spectrum. For its construction, the stop-ingas and the T O F cuts were implemented. The TINA was calibrated on the 129.4 MeV-peak in terms of its hardware sum signal digitised by the ADC-3512. The resulting energy calibration with 7.5% resolution was ETINA —  0.0175 (.CM.3512 — 20.)  (65)  where DA3512 is the digitised ADC word for the ADC-3512. The calibration of the BARS was considerably more involved. It consisted of performing a software balance while implementing a "position-dependent correction."  The output of the A D C for each individual bar (given a fixed amount of deposited energy) depended on the position of the hit along the length of that bar, for the incident particle. With an exponentially decaying pulse-height, a suitable position-dependent "energy-function" has been shown [45] to be Gi = G{POSNi)  = y/ELi • ERi  i = l,---,7  (66)  where: • EL, : ADC (pedestal corrected) output from Left hand of the i bar. t/l  • ER, : ADC (pedestal corrected) output from Right hand of the i bar. </l  • POSN, : POSitioN along the length of the \  t h  bar; taken to be zero at  the geometric centre of the bar. The BARS was placed in an 86 MeV/c backward p  +  beam and the hardware  balance was performed by arranging centre-hits for each bar in turn. Once the BARS was balanced, position-dependent energy measurements were carried out for two of the bars (these two were bar #3 and #5 which were situated symmetrically about the centre bar) by scanning (every 5 cm) along their respective lengths. 67  oo  A  7r°+rK-(Tr + p ) ^ n + 7  -i—i  7+7  <  5?  77 —box  129.4 MeV peak  0  i  20  i  40  1  i  60 80 E (MeV)  1  100  r  120  140  ?  Figure 18: The "ideal" energy spectrum for 7r~p reaction at rest.  2500  Figure 19: Typical TINA observed energy spectrum. 68  Now, in order to be able to correct the G;'s for their position-dependence, we needed a way of evaluating POSN; for a given EL; and ER;. This was achieved by a (weighted) linear-fit to the "single-bar energy-asymmetry function" (EAS;) for all of the centre-hits and the scanning measurements data available. The single-bar energy-asymmetry function was defined as  where, "(• • •)" represents average value. The resulting fit for POSN; was  POSNi = -22.040 EASi + 0.345  i = 1, • • •, 7  (68)  Once again employing (weighted) linear-fits, position-dependence of the G;'s was removed by elimininating deviations from the respective centre-hit values. This correction term was estimated to be  A G , = 22.331 EASi - 0.349  i = l,---,7  (69)  Figure 20 shows the energy-function (G;) with its position-dependence-corrected form (GC;) presented in figure 21. Knowing the energy of the incident u. beam (T + = 30.58 MeV), the individual +  M  bars were then calibrated by estimating the energy deposited in the Nal part of the BARS. This was achieved by computing the amount of energy loss in the scintillator counters, in the casing and the wrappings of BARS. A modified version of the subroutine ELOSS [49] was employed for this means. derived were 69  The bars calibrations so  -10  -5  0  5  POSN, (cm) Figure 20: The position dependence of the energy function G,. The data for both # 3 and # 5 bars.  &B1 &B1 771  (pdc)  = (G — A G i -6.18)0.0810  (70)  = (G -- AG - 10.33) 0.0820  (71)  = (G -AG  (72)  1  2  2  3  -  -AG  4  -8.17)0.0814  (73)  -AG  5  -4.42)0.0804  (74)  6  - 7.03)0.0811  (75)  3  771  (pdc)  &B4 171  (pdc)  ^Bh  =  (<?4  = (G  6  jp(pdc) = (G -AG  &B6  6  3.25)0.0800  = (G -- AG •- 11.01) 0.0822 7  (76)  7  where E^f^ is the position-dependence-corrected (pdc) energy in the i  th  bar. In  this way, the software-balanced energy expression for the BARS as a whole was TTi(pdc)  &BARS —  _ i=l  70  I7i(P (pdc) (  ^Bi  (77)  POSNj (cm) Figure 21: The corrected energy function GC;. The data for both #3 and #5 bars.  Figure 22 shows the raw energy spectrum observed in one of the member elements of the LST. The spectrum is characterised by a exponentially-decaying low energy Compton scattering region and a "1-electron peak" which corresponds to the average energy deposited by one photon-converted electron (0.6 MeV) in the given thickness (0.32 cm) of the scintillator material. All three of LST1, LST2, and LST3 were calibrated based on this 0.6 MeV peak.  IV.4  Discussionof the (NEU-related) Cuts and the EnergyReconstruction  The various cuts employed in the construction of the neutral time spectrum are the subject of discussion in this section. These cuts varied considerably in their degree of effectiveness; the most important one was the "LST cut." A useful survey of these cuts is also found in [39]. The basic (NEU-related) cuts used were 71  • Prompt cut. • Cosmic cut. • 2  nd  //stop cut.  • TINA-BARS timing cut. • 2  nd  TINA cut.  • Neutron removal cut. • Bremsstrahlung cut. The 2 //sec long time-window for observing a // decay had two distinct time regions: a foreground region (985 nsec long) and an early-time background region (1015 nsec long). These two regions were separated by the "prompt peak" (full width ~20 nsec) which signalled the arrival of a // into the target region. The PROmpt (PRO) cut was used to discard events which were prompt-intime with a // arrival. It achieved this by inspecting both the S2 T D C and the SO-fSl pulse separator TDC's. This cut aimed at removing the 7r -related background and -  its associated 43.37 nsec structure. The cosmic (COS) cut removed any event which was preceded by the passage of a cosmic ray through the apparatus (as far back as) 8 //sec prior to the prompt timing for the event. This long veto was necessary to discriminate against cosmic ray /i's which stopped near TINA. It relied on the delayed IN-OUT signal from the IN and the OUT cosmic counters to achieve this purpose. The 2  nd  //stop cut was briefly discussed in section III.5.  It rejected those  events which occurred as a result of there being more than one \i at a time in the target. Since the technique for this experiment was non-stroboscopic, this cut 73  guarded against attenuation of the observed asymmetry due to the possible dephasing caused by the presence of these additional /x's. This cut was implemented by placing the longest possible time separation between the prompt time and the 2 fj, nd  arrival time. It was required that the 2  nd  fx pulse separator TDC's output value to  be greater than the time value set by the edge of the early-time background (namely 2000 nsec); in this way a 1015 nsec time gap was placed between the prompt signal and the 2  nd  \JL event time. The effect of reducing this gap was to place an exponential  decay edge (next to the prompt peak) on an otherwiseflatbackground. The TINA-BARS cut timing was a pre-pile-up cut.  For an event to have  survived this cut, its TINA-BARS timing signal had to fall within a certain narrow timing band (~20 nsec). With the 2  nd  TINA cut, the TINA pulse separator TDC's were studied and  pile-up in TINA was inhibited. The neutron removal cut has already been discussed. By requiring simultaneous energy deposition (>5 MeV) in the BARS and the TINA separately, with a total energy deposition >25 MeV, it was possible to discriminate against the neutron background by a factor of > 100. Bremsstrahlung radiation arising from the interaction of the decay electrons inside the target and/or within the C-magnet pole faces was discriminated against by the Bremsstrahlung (BREM) cut; this was carried out by the study of the S3, S4, and S5 TDC's and their respective ADC's. The Layered ScinTillator (LST) cut was the most successful in removing the background. It was extensively studied and then optimised. By constructing E _ T min  LS  (minimum energy deposited in LST1, LST2, and  LST3 for a given event), it was possible to remove most of the Compton scattered events allowing the clear identification of the photon-converted 1-electron peak. This is shown in figure 23. The 2-electron peak is not visible because of the high 74  energy cosmic ray background. The LST cut amounted to placing a LOw (LO) and a High (HI) energy requirement on the LST's three elements. The "LO (energy) cut" was employed to identify a "firing" in any of the LST1, LST2, or LST3; it set the minimum energy deposition required in the LST before an event was to be accepted. The "HI (energy) cut" discriminated against the high energy cosmic rays passing through the LST. In connection with the LO cut, two options were studied: "All Three (AT) firing," and "At Least One (ALO) firing." The former case required a simultaneous energy deposition above the minimum energy level in all three of the LST's elements while for ALO this was required this for at least one of the elements. The LO cut was optimised by varying the LO value (Keeping the HI value fixed) and searching for a maximum in the integral Signal-to-Noise (S/N) ratio. Table V presents the optimisation results for the AT case with 57.0 < E < 95.0 7  and the HI value set to 1.5 MeV. Table VI shows the same result for the ALO case with the same conditions. In figure 24 we see how S/N varies with the LO value for the cut. With its far better S/N ratio, the AT option was chosen and its LO value set to 0.15 MeV. In the case of the HI cut, we were mostly interested in its effect on the size of the error bars associated with the observed photon asymmetry. The result of the optimisation study with this parameter showed insensitivity to the HI value; we settled for the value of 1.50 MeV which safely included the 2-electron peak (centred at 1.20 MeV.) And now, the problem of "energy reconstruction" is discussed. It was not adequate to add simply the energies in TINA and the BARS and regard it as the total energy of the RMC photon. This energy had to be appropriately adjusted for the energies deposited in the LST and the non-active elements between the BARS 75  o o  CO  o o in  o o  o o  o o  o o  CN  (siiun - q j v ) (  3  )N  Figure 23: Minimum energy deposition spectrum in LST for a given event. Constructed by choosing the minimum energy deposited in LST1, LST2, and LST3 for a given event. 76  Table V: Optimisation results for the "All Three" (AT) option of LOw (LO) cut.  LO value (MeV)  S/N  signal discarded  background discarded  0.06 0.15 0.20 0.30 0.50 0.60 1.00  6.20 7.14 7.37 7.51 6.71 6.14 4.20  61% 61% 62% 62% 71% 78% 92%  96% 96% 96% 96% 97% 98% 99%  Table VI: Optimisation results for the "At Least One" (ALO) option of LOw (LO) cut .  LO value (MeV)  S/N  signal discarded  background discarded  0.06 0.15 0.20 0.30 0.50 0.60 1.00  1.50 1.76 1.88 2.08 2.41 2.47 2.21  42% 45% 47% 49% 53% 55% 71%  73% 78% 80% 83% 86% 87% 91%  77  and the TINA. Each counter in the LST was calibrated (with rather poor resolution) based on the 1-electron peaks in its energy spectra. The energy deposited in the LST in turn provided us with an estimate for the total number of photon-converted electrons traversing between the BARS and TINA; the energy deposited in the nonactive elements which depended on this number was then estimated accordingly from the previous work done with the EGS(3) [50] Monte Carlo programme. The estimated value for this energy was ~2.5 MeV for a single electron and ~5.0 MeV for two electrons passing between the BARS and the TINA as a consequence of photon-conversion in the BARS. The sum of the energies deposited in the BARS, TINA, LST, and the nonactive elements between the BARS and the TINA is the so called "reconstructed energy." This energy was then identified as the "actual" energy of an RMC photon.  IV.5  Neutral Data Analysis  IV.5.1  Background Term  After the application of all of the cuts to the raw neutral data, the background was substantially reduced and appeared quite flat. Any structure which might have remained (due to the cyclotron-related background) was overshadowed by the Poisson uncertainty associated with individual data points. Consequently, a constant background term B(t) = B = const was chosen. This choice was later substantiated by "scanning" the neutral and charged time spectra over time ranges > 43 nsec to look for possible periodicity in the observed asymmetry; this periodicity being indicative of the 43 nsec time structure in the data. 79  IV.5.2  E s t i m a t i o n of the Precession F r e q u e n c y a n d the Initial P h a s e  Because of the abundance of the decay e  data and its associated large observed  +  asymmetry (no depolarisation of /x within the target), the e +  +  time spectrum was  used for (precession) frequency and initial phase determination. Figure 25 shows the e  time spectrum for the "energy slice" 15.0 MeV < E + < 55.0 MeV. The  +  e  corresponding energy spectrum is shown infigure26. The cuts applied for the analysis of the charged data were a subset of the NEU-related cuts. These were: • Prompt cut. • Cosmic cut. • 2  nd  /xstop cut.  • TINA-BARS timing cut. • 2  nd  TINA cut.  • S4 and S5 timing cut. The energy loss within the target (due to the interaction of the e with the material +  of the target) was also accounted for. From previous EGS(3)-related work this was estimated to be ~7.0 MeV for this experiment. From a x —minimisationfitto the e time spectrum by a function of the form 2  +  given by the equation 60, it was possible to estimate the precession frequency and initial phase. To carry out these fits, the e data were "packed at 10 bins" (i.e. a +  "rebinning" of the T D C channels by a factor of 10); with our time calibration this meant a 10.11 nsec "time resolution." Furthermore, the constancy of these fitted values was investigated by performing "time and energy scans." 80  i  0  i  100  200  •  experimental  -  fit  i  i  1  300 400 500 t (nsec)  1  600  :  —i  700  800  Figure 25: e time spectrum with the given energy slice 15 MeV< E <55 MeV. +  e+  50000  1  20  30  1  E  e +  1  40 50 (MeV)  60  70  Figure 26: The experimental energy spectrum for the decay e data. +  81  Table VII: Results of the time scan for the e trum,where the superscript 10 on the rebinning factor.  <t>  tl  (nsec) 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 56.0 64.0 71.0 78.0 85.0 92.0  (psec)  2.1710.02 2.1410.04 2.1510.05 2.1410.06 2.1810.02 2.1910.02 2.1810.02 2.1910.02 2.1710.04 2.1610.02 2.1710.02 2.1610.03 2.1510.04 2.1710.03 2.2110.05  V  (°)  (MHz)  88.011.8 88.010.9 88.011.1 89.011.5 89.010.7 88.010.5 88.011.8 88.010.8 89.011.2 89.011.8 89.011.9 89.011.9 89.012.1 89.012.2 87.012.1  5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01 5.5010.01  +  time specsignifies  A (10) A  e+  0.18310.003 0.18410.003 0.18410.003 0.18210.003 0.18210.003 0.18210.002 0.18310.003 0.18310.003 0.18210.003 0.18210.003 0.18210.003 0.18210.003 0.18210.003 0.18010.003 0.18310.002  Table VII presents the results of the time scan (with the given energy slice 15.0 MeV < E + < 55.0 MeV.) The time ti (measured relative to the prompt time e  t ), signifies 0  the start of the fitting region and is placed at the left-edge of the very  first bin in that region. The decay life time for p from these fits agrees reasonably +  well with the present world average for the p life time:  =2.1970310.00004 psec  [51]. The estimated values of the initial phase angle (4>) also agree quite well with the nominally expected value of 90°. The measured value of the precession frequency (U>/2TT) is 5.5010.01 MHz. Finally, the (integral) observed e  +  asymmetry shows  a remarkable stability for times up to ti ~ 300 nsec; this is shown graphically in figure 27. 82  Figure 27: Stability of the (integral) observed e asymmetry +  83  Table VIII: Results of the energy scan for the e time spectrum. +  E + Range  <f> (°)  e  (MeV) 15.0—• 20.0—• 25.0—• 30.0—> 35.0—• 40.0—>  55.0 55.0 55.0 55.0 55.0 55.0  (//sec) 2.1710.03 2.1610.03 2.1910.05 2.1210.20 2.1510.04 2.1210.03  89.011.5 89.011.7 89.010.8 90.010.9 89.011.4 89.010.7  V  (MHz) 5.5010.01 5.5010.01 5.5010.00 5.5010.01 5.5010.01 5.5010.00  Table IX: // stop statistics for the RMC runs.  Comment  ( . ) =86 MeV/c runs Pfi  //stop  7.37624 x IO  (Pn-) = 76 MeV/c runs  4.70885 x 10  10  The results of the energy scan for the e table VIII; here t  x  +  10  Total statistics  12.08509 x IO  10  time spectrum are shown in the  = 85 nsec. Once again, consistent values for the initial phase  and the frequency were obtained.  IV.5.3  Extraction of the Observed Photon Asymmetry  The RMC asymmetry measurements were carried out at two different average beam momenta: 76 MeV/c and 86 MeV/c. Table IX shows the total nember of //stop for each of these two cases. With the rather limited number of "clean" events available, an independent statistical analysis for these two different momenta separately, was abandoned. Instead, all of the available data at both momenta were summed up to produce one 84  Table X: Optimisation results for E which is the upper limit of the energy slice over which the observed photon asymmetry is defined. max  E (MeV)  S76  S86  7  2395 1521 2862 1785 3094 1931 3217 1989 3249 2012 3252 2019  70.0 75.0 80.0 85.0 90.0 95.0  3916 4647 5025 5206 5261 5271  neutral time spectrum. The observed photon asymmetry was then estimated by a fit to this time spectrum. Figure 28 shows the resulting time spectrum at 1.01 nsec/bin time resolution after the application of all of the cuts. For the neutral time spectrum, the integral signal (S) for the incident average beam momentum (p - ) was defined as M  /•960  J  /-960  I  o  r  r  rO  N(t)dt-  /  0  B(t)dt  (78)  7-960  where A E was the energy slice over which the observed photon asymmetry was 7  defined. Now, as RMC is prominent over photon energies > 57.0 MeV, the obvious choice for the energy slice became A E  7  = E  max  — 57.0. The value of E  m a x  was then  chosen so as to maximise the signal. The result of this optimisation is presented in table X. From the above optimisation it became clear that there was no real gain in the RMC signal for E  7  > 95.0 MeV. The (integral) observed photon asymmetry 85  Figure 28: Final neutral time spectrum shown at the collected 1.011 nsec time resolution showing both the background and the foreground time regions. 86  was then appropriately estimated as Ay=  /  A {Ey)dE =  Ay(E )dE  1  1  1  1  (79)  With the limited statistics in hand, a fit to the data at the best time resolution of 1.01 nsec/bin was impractical and unsuitable. Now, in order to reduce the Poisson uncertainty associated with individual data points used in the fit, it was necessary to rebin and fit the data at coarser time resolutions. The resulting value of the observed photon asymmetry extracted from the neutral time spectrum was then somewhat attenuated by the ("angular") averaging brought about by this rebinning of the data. If T is the precession period, and Tf, is the "elementary" bin time ([T;,] is the p  corresponding rebinning factor) used in the fit, this attenuation effect is expressible in terms of a reduction factor (rpy) given by  where the "rebinning-associated angle" 9[T ] is b  %y = ^ x  (81)  2TT  If A^Td) is the estimated value of the attenuated observed photon asymmetry from the fit, then the true value of this asymmetry, taking the rebinning into account, is given by A  y  =  (82)  In this experiment, the precession frequency of 5.50 MHz corresponded to T = 181.8 nsec resulting in r p  1 0  = 0.995 and r  4 3  = 0.911.  In relation to the \ ~minimisation fit procedure, the following is true: 2  • By a Semi-Free" (SF) fit we mean one in which all of the parameters as u  per equation 60, except the background term (B is fixed at its experimentally determined value), are allowed to vary freely in the x minimisation process. 2—  87  • On the other hand, a "Partially-Restricted" (PR) fit is one for which u, <f>, and B are fixed at their experimentally determined value and r is fixed at 333.0 nsec ref [18].  Figure 29 shows the neutral time spectrum rebinned by a factor of 43 (a time resolution of 43.37 nsec) with its associated extracted and attenuated observed photon asymmetry shown in figure 30. The corresponding plots at the higher time resolution of 10.11 nsec are shown infigures31 and 32. In both cases, the fits shown are semi-free fits. In order to justify the choice of a flat background, a time scan of the observed photon asymmetry was carried out. The result of this scan is shown in figure 33. The scan was sufficiently long to detect any possible periodicity existing in the time spectrum due to the 43.37 nsec time structure associated with the cyclotron-related background. Furthermore, it was necessary to ensure that rebinning by an arbitrary factor did not artificially affect the extracted value of the observed photon asymmetry. This was achieved by comparing results of time scans for two different rebinningfactors: 43 and 10.  The reader is reminded that t  x  is the starting time for the  fitting region. It is noted that that with a rebinning factor of 43 (time resolution 43.47 nsec), the background could be genuinely regarded as flat. This is because our time resolution of 43.47 nsec matches closely the 43.37 nsec cyclotron-related time structure. Table XI and figure 34 present the result for the comparison for the two different binning factors; the fits are SF fits. This study provided evidence in favour of insensitivity (of the extracted value of the observed photon asymmetry) to the value of the rebinning factor. 88  600 57.0 < E < 95.0  500  7  w  • -  ^400  experimental fit  fi300  s g200H 100 0  0  200  400 600 t (nsec)  800  1000  Figure 29: Neutral time spectrum with the time resolution of 43.47 nsec; ti = 40 nsec.  400 600 t (nsec)  1000  Figure 30: Attenuated observed photon asymmetry A^ ). 43  89  n  0  1  200  1  1  400 600 t (nsec)  r  1  800  1000  Figure 31: Neutral time spectrum with the time resolution of 10.11 nsec; t = 40 nsec. x  -0.5-  200  400 600 t (nsec)  800  Figure 32: Attenuated observed photon asymmetry 90  1000 A„ \ w  F i g u r e 33: T i m e scan of the attenuated observed asymmetry at 43.47 nsec time resolution.  91  photon  o LO  i  L O  1—i  a—•  1  H-  o  I—I  o  M  B-«  LO  o o  43  E-i  o  I  2  X>  EH  CD  1  •  B  -H  1  1  ro  1  LO  1  CN  O CN CN  CO O  CN  L  o  O O  V  Figure 34: Comparison of the time scans for the observed photon asymmetry with 43.47 nsec and 10.11 nsec time resolutions. 92  W  Table XI: Table of the time scans for the observed photon asymmetry with 43.47 nsec and 10.11 nsec time resolutions and semi-free (SF) fits.  tl (nsec)  A^ —  ,(io) 1  A  no  Xv  A-y —  ,(«) A  xl  ~<  25.0  0.127  +0.0230 -0.0233  86.9 90  = 0.97  0.115  +0.0247 -0.0246  21.9 16  = 1.37  29.0  0.116  +0.0232 -0.0234  86.3 90  = 0.96  0.102  +0.0243 -0.0252  23.4 16  = 1.46  33.0  0.112  +0.0239 -0.0236  89.3 89  = 1.00  0.113  +0.0248 -0.0252  22.2 16  = 1.39  37.0  0.117  +0.0247 -0.0237  84.8 88  = 0.96  0.123  +0.0260 -0.0259  16.8 16  = 1.05  41.0  0.112  +0.0251 -0.0247  102.6 88  = 1.17  0.113  +0.0265 -0.0263  11.1 15  = 0.74  45.0  0.103  +0.0255 -0.0249  79.3 88  = 0.90  0.112  +0.0277 -0.0275  8.9 15  = 0.59  The final chosen value of the observed photon asymmetry is the one obtained from a PR fit to the neutral time spectrum at 43.47 nsec time resolution. The fitting region was chosen with the smallest possible value of ti = 25 nsec; as this choice provides the largest amount of statistical information. Table XII presents the results of the PR and the SF fits to the neutral time spectrum for ti = 25 nsec.  ^  AW = ^  0.101010.0213 , „ = ,0.911 =°^ n  +  1  93  1  1  n  n  n  0  2  n 3  ( ) 83  Table XII: Final PR and SF fits to the neutral time spectrum; the region of fit starts at ti = 25 nsec with 43.47 nsec time resolution. Type of fit  Parameter  Value  A< )  0.10110.0213  43  7  PR  T  333.0 (nsec)  V  5.50 (MHz)  <t>  90 (degrees) 25.5/16 = 1.59  xl  A > 43  7  SF  U.1UU_  0  0  224  r  339.015.1 (nsec)  V  5.5710.08 (MHz) 103111 (degrees) 21.9/16 = 1.37  xl  94  Chapter V Discussion of Results, a n d Conclusions  V.l  Brief Review of Some Aspects of Data Analysis  Before presenting our value for  and discussing similar measurements, it is  worthwhile to briefly review the major features in the data analysis which aimed at ensuring a consistent and meaningful extraction of the observed photon, asymmetry, Ay. Due to the limited size of the data set, the neutral time spectrum had to be rebinned in order to allow meaningful fits. Consequently, the first task was to ensure that the extracted value of A was not affected artificially or systematically 7  by the bin-size chosen. This was substantiated by performing a "time scan" (i.e. varying the starting time for the fitting region relative to the prompt time) at two different time resolutions of 10.11 nsec and 43.47 nsec. The two time scans agreed within the statistical uncertainties associated with the data. The final value was obtained from a fit with 43.47 nsec time resolution, for which the background could be genuinely regarded as flat. The much improved data quality (signal to noise ratio of ~7) meant a much reduced dependence on any possible systematics associated with a backgroud-subtraction carried out on the neutral time spectrum. 95  V.2  D e t e r m i n a t i o n o fa  7  By normalising the value of the observed photon asymmetry to the residual y~ polarisation, the value for the physical photon asymmetry becomes ^ 7  =  011110023  =  (Pp-) o.inio.oio  ~  v  ;  the fit has a xl value of 1.59 which is quite acceptable. It is noted that a value of exactly +1 for the photon asymmetry implies a pure V—A interaction [52,53] (this corresponds to a complete suppression of the induced coupling constants). Even though the final result for A has been obtained from a "constrained" 7  fit, the corresponding "unconstrained" fit (with xl  =  1-37) provides an A in very 7  good agreement. Furthermore, the unconstrained fit produces an acceptable value for the //"-decay life time calcium: r - = 339J5 nsec, and is able to "find" the M  asymmetry signal (by reproducing the expected values of the initial phase and the precession frequency within the statistical uncertainties associated with the data.) Table XIII presents all the results of measurements of the photon asymmetry for  4 0  Ca. Our present value is in agreement with all values except the one reported  by DiLeila et al. DiLella et al. employed a "bare" Nal detector without any converter. Their estimated background due to the inclusion of OMC neutrons was ~45%. Their result is the only negative value obtained experimentally for the photon asymmetry and it is in sharp disagreement with the theories  [5,27,60] which predict a large  and positive photon asymmetry. Subsequent to that work, all the later experiments implemented an OMC-neutron-discriminating-capability in their set-up by placing a converter (active or non-active) in front of their main Nal detector. The result of Hart et al. was obtained by a constrained fit (xl ~1.0) to their time spectrum containing ~1200 events. With ~30% energy resolution, they chose 63.5 MeV as their lower cut-off value for the RMC photon energy spectrum. From 96  Table XIII: Summary of photon asymmetry measurements on C a . \ is the preliminary result based on part of the data analysed by Virtue. The world average quoted excludes the DiLella et al. result; its error has been estimated by the "quadrature rule" for statistically independent measurements. 40  dy  Reference  < -0.32i0.48 0.9010.50 0.82±0.76 0.9910.43 i 0.9210.26 0.9410.14 1.0010.23  DiLella et al. 1971 [54] Hart et al. 1977 [55] Frischknecht et al. 1980 [56] Hasinoff et al. 1984 [57] Dobelli et al. 1986 [58] Schaad 1987 [59] Virtue 1987 [39] Present work  0.9810.17  The world average  1 OO+0.54  their e -data they obtained a value of r^- = (36618) nsec which is considerably larger than the presently accepted value : r^- ~333 nsec [18]. Frischknecht et al., Dobelli et al., and Schaad present the results of the SIN group. They employed a pair-spectrometer with the stroboscopic method. (In this method, the muon precession frequency is tuned to that of the cyclotron frequency.) Their latest value for a  is that reported by Schaad which is extracted from a  y  constrained fit to a time spectrum containing some 40000 events. The author does not quote the xl f ° his final fit. From their graphical presentation, we estimate a r  value of xl ~4; a large value which makes the quality of their fit somewhat poor. Hasinoff et al. and Virtue presented the results for the experiment from which the present work developed. The value given by Virtue is to supercede the earlier and preliminary value quoted by Hasinoff et al. Virtue's result was obtained by an unconstrained fit (xl  =  1-10) to a time spectrum containing 2500 events. His 97  value for r - as determined from the decay-e time spectrum was also larger than -  M  expected: r - = (352^4) nsec. This discrepancy remains unresolved. For ti < M  67 nsec, the extracted value of A showed signs of systematic problems implying 7  some distortion in the time spectrum for these early times. His final fit was chosen for i i < 67 nsec to avoid the initial problematic region in the time spectrum. In view of our present result, we are inclined to explain his observed distortion in statistical terms, although Hart et al. were also forced to discard the first 100 nsec following the \x~ stop. Vitrue's value of the photon asymmetry, even though it is unphysical (the magnitude of a can not be larger than 1), is in agreement, within 7  the statistical uncertainties, with our present value and the other results. (Once again excluding the value of DiLella et al.) We had hoped for 10000 clean RMC photon events to provide a 10% error in the photon asymmetry. Due to insufficient allocation of beam time, only half as many RMC photons were collected; resulting in ~20% uncertainty in the extracted value for the photon asymmetry.  V.3  E x t r a c t i o n of  gp/gA  The method employed for extracting gp/gA from the experimental photon asymmetry is a graphical one and relies on the almost model independence of the photon asymmetry. Figure 35 shows a as a function of gp/gA for the Christillin model [32] 7  (which uses a phenomenological nuclear response function made up of a dipole and a quadrupole part) and also for the model of Gmitro et al. [6] (which employs detailed wavefunctions for the initial and all important final nuclear states, i.e. it performs a detailed microscopic calculation summed over all the partial transitions.) Our extracted value will be based on the Christillin model with the slightly larger error 98  bound; the resulting value is 9P = (0-0t ° )9A  (85)  8  6  7  With this given level of accuracy, it is not possible to make a very meaningful statement about the renormalisation of gp. Our result tends to indicate a complete renormalisation (i.e. a full quenching) of gp inside the nucleus of C a while at the 40  same time, with its associated uncertainties, does not exclude the unrenormalised Goldberger-Treiman value [7] of gp « 7gAWe will now discuss some recent results for gp based on the RMC method. Virtue [39] produced two values for gp: One based on a photon asymmetry measurement and the other was derived from his rate measurements. Due to his large value for the photon asymmetry, he could only quote an upper limit for the pseudoscalar coupling constant: gp < 5.4gA', favouring a renormalisation. The model-dependent values o£ gp extracted from his rate measuerment were based on the models of Christillin [32] and Gmitro et al. [33]. His overall value so derived was gp ~ (7^2) gAThe recent SIN results are equally as indeterminate with respect to the question of the renormalistion of gp within the nucleus of C a . They too, present results 40  based on both the photon asymmetry measurements and the partial branching ratio measurements. They derive their model-dependent extracted values of gp based on Gmitro's modified impulse approximation (MIA) method [33,61]. The modified impulse approximation was introduced to remove the discrepancy between the photon yield as predicted by the phenomenological model of Christillin [32] and the earlier impulse approximaiton (IA) of Gmitro et al. [6]. Although the MIA is quite successful in predicting RMC rates, it seems to predict a lesser sensitivity of a to 7  gp and is in sharp departure from the "model independence of gp" as predicted by earlier theories; this matter has been raised and as yet remains unresolved. Schaad's value [59] based on the photon asymmetry measurement is 100  Table XIV: Values of gp/gA as deduced from OMC experiments on hydrogen. The values quoted are from the analysis of Bardin et al. [62]. Target  Reference Bleser et al. 1962 [63] Rothberg et al. 1963 [64] Alberigi Quaranta et al. 1969 [65] Bystritiskii et al. 1974 [66] Bardin et al. 1981 [67] Average  9P/9A  liquid H2 6.0±8.0 liquid H2 11.0±4.3 10.3±3.9 gas H 7.9±5.9 gas H liquid H2 7.1±3.0 8.7±1.9 2  2  gp = (5-lt '.o) gA', once again the problem of renormalisation of gp in C a re4 0  7  mains open. Dobelli et al. [58] report their results for RMC rate measurements (and the corresponding gp values) on several nuclei in the range 12 < Z < 83; their value of gp for C a is gp = (6.311;°) 9A- Their investigation tends to suggest a 4 0  gradual decrease in gp with increasing Z value with a full quenching (i.e. complete renormalisation) of gp at at Z~80. This section is concluded by a brief mention of other alternative methods available for determinimg gp/gA in various nuclei. First we review the situation for hydrogen. Table XIV presents the results of OMC measurements and the values obtained for gp/gA are in agreement with the Goldberger-Treiman value. The experimental results for heavier nuclei are however more confusing. The use of polarised beams in muon capture experiments provides many observables that have some sensitivity to gp, but in OMC, the final state particles are just neutrons and neutrinos and so the measurement of most of these observables is impossible. As a result, the techniques employed in conjunction with the OMC method are not generally applicable to a wide range of nuclei. In fact, only three complex nuclei have been studied by three different techniques. Table X V summarises the results of OMC measurements on complex nuclei used for extracting gp/gA101  Table XV: Values of gp/gA extracted from OMC experiments on complex nuclei. It is assumed that gr — 0. In this table VAV and Px, refer to the average and longitudinal polarisation of the recoiling nucleus respectively; is the capture rate for pT and A/, stands for the /? decay rate. Finally, 7 — 1/ signifies 7 — 1/ a  s  Reference  Method  Possoz et al. 1974 [68] Holstein 1976 [69] Holstein 1976 [69] Possoz et al. 1977 [70] Parthasarathy and Sridhar 1979 [71] Roesch et al. 1981 [72] Kuno et al. 1984,1986 [73,74] Gagliardi et al. 1982 [75] Heath and Garvey 1985 [76] Hamel et al. 1985 [77] Ciechanowicz 1976 [78] Parthasarathy and Sridhar 1981 [79]  P'AV PAV  K  PAV PAV  PAV  A/?A A/jA^ A/jA^ 7 — 1/ 7 — 1/ M  Nucleus 12  C  12  C  12  C  12  C  12  C  12  C  12  C  16Q 16Q 16Q 28 28  S i  Si  9P/9A  12±5 15±4 8.5±2.5 7.1±2.7 13.3±1.8 9.4±1.7 10.1±2.4 10±2.5 « 12.5 11 - 12 -1.9±3.1 13.5±f:|  We see that results are somewhat incompatible and tend to vary as further improvements are made in the intrepretaion. Often there are rather subtle nuclear structure effects which have to be taken into account and the systematic errors are thus hard to estimate. The overall impression is that the value of gp is enhanced not quenched; but this is the opposite of the trend of the resutls for RMC. It is therefore essential that further experimental and theoretical studies be carried out to determine the answer to this conundrum.  V.4 Conclusions For this second TRIUMF experiment, the major sources of background had already been identified. 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